The groups of order 64
The groups of order 64
This page was prepared by GAP.
It contains information about the groups of order 64, as
- presentation (PC-representation)
- order distribution of the elements
- number of conjugacy classes
- nilpotency class
- centre and commutator subgroup
- automorphism and inner automorphism group
- lower central series
For the groups 64/261 - 64/267 no information about their automorphism group
is given. This information could not be computed in reasonable time.
The groups are numbered according to their numbering in GAP.
64/1
((a1,a2,a3,a4,a5,a6;a1^2=a2,a2^2=a3,a3^2=a4,a4^2=a5,a5^2=a6,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
1 elements of order 2
2 elements of order 4
4 elements of order 8
8 elements of order 16
16 elements of order 32
32 elements of order 64
ABELIAN
--------------------------------------------------------------
64/2
((a1,a2,a3,a4,a5,a6;a1^2=a3,a2^2=a4,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
12 elements of order 4
48 elements of order 8
ABELIAN
group is isomorphic to 8/1 x 8/1
--------------------------------------------------------------
64/3
((a1,a2,a3,a4,a5,a6;a1^2=a3,a2^2=a4,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
12 elements of order 4
48 elements of order 8
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/3
Commutator subgroup generated by [ a5 ] , type 2/1
Automorphism Group of size 512
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/3 --- 2/1 --- 1/1
--------------------------------------------------------------
64/4
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a3,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
16 elements of order 4
32 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5 ] , type 4/2
Automorphism Group of size 256
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/4 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/5
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a3,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5 ] , type 4/2
Automorphism Group of size 256
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/5 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/6
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a3*a5,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
20 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a3, a5 ] , type 4/1
Automorphism Group of size 128
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/6 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/7
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a3*a5,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
28 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a3, a5 ] , type 4/1
Automorphism Group of size 128
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/7 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/8
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a5*a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/2
Automorphism Group of size 128
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/8 --- 8/2 --- 4/2 --- 1/1
--------------------------------------------------------------
64/9
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a5*a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/2
Automorphism Group of size 128
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/9 --- 8/2 --- 4/2 --- 1/1
--------------------------------------------------------------
64/10
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=a6,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a5*a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
20 elements of order 4
32 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/10 --- 8/2 --- 4/2 --- 1/1
--------------------------------------------------------------
64/11
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=a6,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a5*a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
28 elements of order 4
32 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/11 --- 8/2 --- 4/2 --- 1/1
--------------------------------------------------------------
64/12
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=a6,a4^2=a5*a6,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a5*a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
12 elements of order 4
32 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/12 --- 8/2 --- 4/2 --- 1/1
--------------------------------------------------------------
64/13
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=a6,a4^2=a5*a6,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a5*a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
28 elements of order 4
32 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/13 --- 8/2 --- 4/2 --- 1/1
--------------------------------------------------------------
64/14
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a6,a3^2=a6,a4^2=a5*a6,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a5*a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
28 elements of order 4
32 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/14 --- 8/2 --- 4/2 --- 1/1
--------------------------------------------------------------
64/15
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a3,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a3*a5,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
12 elements of order 4
48 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a3, a5 ] , type 4/1
Automorphism Group of size 256
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/15 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/16
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a3*a5,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a3*a5,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
12 elements of order 4
48 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a3, a5 ] , type 4/1
Automorphism Group of size 256
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/16 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/17
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a3,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a3 ] , type 2/1
Automorphism Group of size 512
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/17 --- 2/1 --- 1/1
--------------------------------------------------------------
64/18
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a3, a6 ] , type 4/1
Commutator subgroup generated by [ a3, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 16/3
Nilpotency class 2
Lower Central Series: 64/18 --- 4/1 --- 1/1
--------------------------------------------------------------
64/19
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=a6,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
8 elements of order 4
48 elements of order 8
22 conjugacy classes
Centre generated by [ a3, a6 ] , type 4/1
Commutator subgroup generated by [ a3, a6 ] , type 4/1
Automorphism Group of size 1536
Inner automorphism group type 16/3
Nilpotency class 2
Lower Central Series: 64/19 --- 4/1 --- 1/1
--------------------------------------------------------------
64/20
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3*a5, a4, a6 ] , type 8/2
Commutator subgroup generated by [ a3, a6 ] , type 4/1
Automorphism Group of size 256
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/20 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/21
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3*a5, a4, a6 ] , type 8/3
Commutator subgroup generated by [ a3, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/21 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/22
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=a6,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
8 elements of order 4
48 elements of order 8
28 conjugacy classes
Centre generated by [ a3*a5, a4, a6 ] , type 8/2
Commutator subgroup generated by [ a3, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/22 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/23
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a3,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/23 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/24
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a3,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/24 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/25
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a3,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/1
Commutator subgroup generated by [ a3, a6 ] , type 4/2
Automorphism Group of size 256
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/25 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/26
((a1,a2,a3,a4,a5,a6;a1^2=a3,a2^2=a4,a3^2=a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
12 elements of order 4
16 elements of order 8
32 elements of order 16
ABELIAN
group is isomorphic to 4/1 x 16/1
--------------------------------------------------------------
64/27
((a1,a2,a3,a4,a5,a6;a1^2=a3,a2^2=a4,a3^2=a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
12 elements of order 4
16 elements of order 8
32 elements of order 16
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/2
Commutator subgroup generated by [ a6 ] , type 2/1
Automorphism Group of size 256
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/27 --- 2/1 --- 1/1
--------------------------------------------------------------
64/28
((a1,a2,a3,a4,a5,a6;a1^2=a3,a2^2=a4,a3^2=a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a5*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
12 elements of order 4
16 elements of order 8
32 elements of order 16
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/1
Commutator subgroup generated by [ a5, a6 ] , type 4/1
Automorphism Group of size 128
Inner automorphism group type 16/3
Nilpotency class 2
Lower Central Series: 64/28 --- 4/1 --- 1/1
--------------------------------------------------------------
64/29
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=1,a4^2=a5,a5^2=a6,a6^2=1
,[a1,a2]=a3,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
8 elements of order 4
16 elements of order 8
32 elements of order 16
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/2
Commutator subgroup generated by [ a3 ] , type 2/1
Automorphism Group of size 128
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/29 --- 2/1 --- 1/1
--------------------------------------------------------------
64/30
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=1,a4^2=a5,a5^2=a6,a6^2=1
,[a1,a2]=a3,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
8 elements of order 4
16 elements of order 8
32 elements of order 16
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/1
Commutator subgroup generated by [ a3, a6 ] , type 4/2
Automorphism Group of size 128
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/30 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/31
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=a6,a4^2=a5,a5^2=a6,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
8 elements of order 4
16 elements of order 8
32 elements of order 16
28 conjugacy classes
Centre generated by [ a3*a4, a5, a6 ] , type 8/1
Commutator subgroup generated by [ a3, a6 ] , type 4/1
Automorphism Group of size 64
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/31 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/32
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a3,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
28 elements of order 4
16 elements of order 8
13 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/3
Automorphism Group of size 128
Inner automorphism group type 32/46
Nilpotency class 4
Lower Central Series: 64/32 --- 8/3 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/33
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a6,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a3,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
13 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/3
Automorphism Group of size 128
Inner automorphism group type 32/46
Nilpotency class 4
Lower Central Series: 64/33 --- 8/3 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/34
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a5*a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
44 elements of order 4
13 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 32/46
Nilpotency class 4
Lower Central Series: 64/34 --- 8/2 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/35
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a6,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a5*a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
52 elements of order 4
13 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 32/46
Nilpotency class 4
Lower Central Series: 64/35 --- 8/2 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/36
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a5*a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
20 elements of order 4
32 elements of order 8
13 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 32/46
Nilpotency class 4
Lower Central Series: 64/36 --- 8/2 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/37
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a6,a3^2=a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a5*a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
28 elements of order 4
32 elements of order 8
13 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 32/46
Nilpotency class 4
Lower Central Series: 64/37 --- 8/2 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/38
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=a5*a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a3*a5,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a5*a6,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
20 elements of order 4
8 elements of order 8
16 elements of order 16
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/1
Automorphism Group of size 256
Inner automorphism group type 16/12
Nilpotency class 4
Lower Central Series: 64/38 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/39
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a6,a3^2=a5*a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a3*a5,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a5*a6,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
36 elements of order 4
8 elements of order 8
16 elements of order 16
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/1
Automorphism Group of size 256
Inner automorphism group type 16/12
Nilpotency class 4
Lower Central Series: 64/39 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/40
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=a5*a6,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a3*a5,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a5*a6,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
12 elements of order 4
24 elements of order 8
16 elements of order 16
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/1
Automorphism Group of size 128
Inner automorphism group type 16/12
Nilpotency class 4
Lower Central Series: 64/40 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/41
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a3*a5*a6,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
28 elements of order 4
8 elements of order 8
16 elements of order 16
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/1
Automorphism Group of size 128
Inner automorphism group type 32/27
Nilpotency class 4
Lower Central Series: 64/41 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/42
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=a5,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a3*a5*a6,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
4 elements of order 4
24 elements of order 8
16 elements of order 16
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/1
Automorphism Group of size 256
Inner automorphism group type 32/27
Nilpotency class 4
Lower Central Series: 64/42 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/43
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a6,a3^2=a5,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a3*a5*a6,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
20 elements of order 4
24 elements of order 8
16 elements of order 16
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/1
Automorphism Group of size 256
Inner automorphism group type 32/27
Nilpotency class 4
Lower Central Series: 64/43 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/44
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a3,a3^2=1,a4^2=a5,a5^2=a6,a6^2=1
,[a1,a2]=a3,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
12 elements of order 4
16 elements of order 8
32 elements of order 16
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/2
Commutator subgroup generated by [ a3 ] , type 2/1
Automorphism Group of size 128
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/44 --- 2/1 --- 1/1
--------------------------------------------------------------
64/45
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a3,a3^2=a6,a4^2=a5,a5^2=a6,a6^2=1
,[a1,a2]=a3*a6,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
12 elements of order 4
16 elements of order 8
32 elements of order 16
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/1
Commutator subgroup generated by [ a3, a6 ] , type 4/1
Automorphism Group of size 128
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/45 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/46
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a3,a3^2=a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a3*a5*a6,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
20 elements of order 4
24 elements of order 8
16 elements of order 16
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/1
Automorphism Group of size 128
Inner automorphism group type 32/30
Nilpotency class 4
Lower Central Series: 64/46 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/47
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a3*a5,a3^2=a5*a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a3*a5,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
36 elements of order 4
8 elements of order 8
16 elements of order 16
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/1
Automorphism Group of size 512
Inner automorphism group type 16/12
Nilpotency class 4
Lower Central Series: 64/47 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/48
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a3*a5*a6,a3^2=a5*a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a3*a5,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
36 elements of order 4
8 elements of order 8
16 elements of order 16
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/1
Automorphism Group of size 512
Inner automorphism group type 16/12
Nilpotency class 4
Lower Central Series: 64/48 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/49
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a3*a5,a3^2=a5*a6,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a3*a5,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
4 elements of order 4
40 elements of order 8
16 elements of order 16
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/1
Commutator subgroup generated by [ a3, a5, a6 ] , type 8/1
Automorphism Group of size 256
Inner automorphism group type 16/12
Nilpotency class 4
Lower Central Series: 64/49 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/50
((a1,a2,a3,a4,a5,a6;a1^2=a3,a2^2=1,a3^2=a4,a4^2=a5,a5^2=a6,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
4 elements of order 4
8 elements of order 8
16 elements of order 16
32 elements of order 32
ABELIAN
group is isomorphic to 2/1 x 32/1
--------------------------------------------------------------
64/51
((a1,a2,a3,a4,a5,a6;a1^2=a3,a2^2=1,a3^2=a4,a4^2=a5,a5^2=a6,a6^2=1
,[a1,a2]=a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
4 elements of order 4
8 elements of order 8
16 elements of order 16
32 elements of order 32
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/1
Commutator subgroup generated by [ a6 ] , type 2/1
Automorphism Group of size 64
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/51 --- 2/1 --- 1/1
--------------------------------------------------------------
64/52
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a4*a5,a4^2=a5*a6,a5^2=a6,a6^2=1
,[a1,a2]=a3*a4,[a1,a3]=a4*a5,[a1,a4]=a5*a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a4*a5,[a2,a4]=a5*a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
33 elements of order 2
2 elements of order 4
4 elements of order 8
8 elements of order 16
16 elements of order 32
19 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a4, a5, a6 ] , type 16/1
Automorphism Group of size 512
Inner automorphism group type 32/49
Nilpotency class 5
Lower Central Series: 64/52 --- 16/1 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/53
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=a4*a5,a4^2=a5*a6,a5^2=a6,a6^2=1
,[a1,a2]=a3*a4,[a1,a3]=a4*a5,[a1,a4]=a5*a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a4*a5,[a2,a4]=a5*a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
17 elements of order 2
18 elements of order 4
4 elements of order 8
8 elements of order 16
16 elements of order 32
19 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a4, a5, a6 ] , type 16/1
Automorphism Group of size 256
Inner automorphism group type 32/49
Nilpotency class 5
Lower Central Series: 64/53 --- 16/1 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/54
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a6,a3^2=a4*a5,a4^2=a5*a6,a5^2=a6,a6^2=1
,[a1,a2]=a3*a4,[a1,a3]=a4*a5,[a1,a4]=a5*a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a4*a5,[a2,a4]=a5*a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
1 elements of order 2
34 elements of order 4
4 elements of order 8
8 elements of order 16
16 elements of order 32
19 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a3, a4, a5, a6 ] , type 16/1
Automorphism Group of size 512
Inner automorphism group type 32/49
Nilpotency class 5
Lower Central Series: 64/54 --- 16/1 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/55
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
ABELIAN
group is isomorphic to 4/1 x 4/1 x 4/1
--------------------------------------------------------------
64/56
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/5
Commutator subgroup generated by [ a4 ] , type 2/1
Automorphism Group of size 12288
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/56 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/18
--------------------------------------------------------------
64/57
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a4 ] , type 2/1
Automorphism Group of size 12288
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/57 --- 2/1 --- 1/1
--------------------------------------------------------------
64/58
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a4 ] , type 2/1
Automorphism Group of size 2048
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/58 --- 2/1 --- 1/1
group is isomorphic to 4/1 x 16/9
--------------------------------------------------------------
64/59
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=a4*a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a4 ] , type 2/1
Automorphism Group of size 2048
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/59 --- 2/1 --- 1/1
group is isomorphic to 4/1 x 16/10
--------------------------------------------------------------
64/60
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
32 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 12288
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/60 --- 4/2 --- 1/1
--------------------------------------------------------------
64/61
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/61 --- 4/2 --- 1/1
--------------------------------------------------------------
64/62
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 4096
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/62 --- 4/2 --- 1/1
--------------------------------------------------------------
64/63
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a5,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 4096
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/63 --- 4/2 --- 1/1
--------------------------------------------------------------
64/64
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4*a5,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 6144
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/64 --- 4/2 --- 1/1
--------------------------------------------------------------
64/65
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 12288
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/65 --- 4/2 --- 1/1
--------------------------------------------------------------
64/66
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a6,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/66 --- 4/2 --- 1/1
--------------------------------------------------------------
64/67
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4*a6,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/67 --- 4/2 --- 1/1
--------------------------------------------------------------
64/68
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a6,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/68 --- 4/2 --- 1/1
--------------------------------------------------------------
64/69
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4*a6,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/69 --- 4/2 --- 1/1
--------------------------------------------------------------
64/70
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a6,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/70 --- 4/2 --- 1/1
--------------------------------------------------------------
64/71
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4*a6,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/71 --- 4/2 --- 1/1
--------------------------------------------------------------
64/72
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4*a5*a6,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/72 --- 4/2 --- 1/1
--------------------------------------------------------------
64/73
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
32 elements of order 4
22 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 3072
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/73 --- 8/3 --- 1/1
--------------------------------------------------------------
64/74
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
22 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 3072
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/74 --- 8/3 --- 1/1
--------------------------------------------------------------
64/75
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
22 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 1024
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/75 --- 8/3 --- 1/1
--------------------------------------------------------------
64/76
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
22 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 3072
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/76 --- 8/3 --- 1/1
--------------------------------------------------------------
64/77
((a1,a2,a3,a4,a5,a6;a1^2=a4*a5,a2^2=a4,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
22 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 1536
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/77 --- 8/3 --- 1/1
--------------------------------------------------------------
64/78
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
22 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 1024
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/78 --- 8/3 --- 1/1
--------------------------------------------------------------
64/79
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
22 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 1024
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/79 --- 8/3 --- 1/1
--------------------------------------------------------------
64/80
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4*a5,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
22 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 3072
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/80 --- 8/3 --- 1/1
--------------------------------------------------------------
64/81
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4*a5,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
22 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 1024
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/81 --- 8/3 --- 1/1
--------------------------------------------------------------
64/82
((a1,a2,a3,a4,a5,a6;a1^2=a4*a5*a6,a2^2=a4*a5,a3^2=a4,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
22 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 10752
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/82 --- 8/3 --- 1/1
--------------------------------------------------------------
64/83
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
ABELIAN
group is isomorphic to 2/1 x 4/1 x 8/1
--------------------------------------------------------------
64/84
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a6 ] , type 2/1
Automorphism Group of size 2048
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/84 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/19
--------------------------------------------------------------
64/85
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=1,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
40 conjugacy classes
Centre generated by [ a2, a4, a5, a6 ] , type 16/3
Commutator subgroup generated by [ a6 ] , type 2/1
Automorphism Group of size 1024
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/85 --- 2/1 --- 1/1
group is isomorphic to 4/1 x 16/11
--------------------------------------------------------------
64/86
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a5,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
40 conjugacy classes
Centre generated by [ a1, a4, a5, a6 ] , type 16/2
Commutator subgroup generated by [ a6 ] , type 2/1
Automorphism Group of size 512
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/86 --- 2/1 --- 1/1
--------------------------------------------------------------
64/87
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
16 elements of order 4
32 elements of order 8
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a4 ] , type 2/1
Automorphism Group of size 1024
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/87 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/20
--------------------------------------------------------------
64/88
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
16 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/88 --- 4/2 --- 1/1
--------------------------------------------------------------
64/89
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
16 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/89 --- 4/2 --- 1/1
--------------------------------------------------------------
64/90
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
22 conjugacy classes
Centre generated by [ a3, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/90 --- 4/2 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/46
--------------------------------------------------------------
64/91
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
22 conjugacy classes
Centre generated by [ a3, a6 ] , type 4/1
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/91 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/92
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
8 elements of order 4
32 elements of order 8
22 conjugacy classes
Centre generated by [ a3, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/92 --- 4/2 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/47
--------------------------------------------------------------
64/93
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
22 conjugacy classes
Centre generated by [ a3, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/93 --- 4/2 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/48
--------------------------------------------------------------
64/94
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
16 elements of order 4
32 elements of order 8
22 conjugacy classes
Centre generated by [ a3, a6 ] , type 4/1
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/9
Nilpotency class 3
Lower Central Series: 64/94 --- 4/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/95
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
24 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 1024
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/95 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/27
--------------------------------------------------------------
64/96
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 1024
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/96 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/28
--------------------------------------------------------------
64/97
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/97 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/98
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/98 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/99
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
24 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/99 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/100
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/100 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/101
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 256
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/101 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/31
--------------------------------------------------------------
64/102
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/1
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/102 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/103
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a4 ] , type 2/1
Automorphism Group of size 1024
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/103 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/21
--------------------------------------------------------------
64/104
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/104 --- 4/2 --- 1/1
--------------------------------------------------------------
64/105
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/105 --- 4/2 --- 1/1
--------------------------------------------------------------
64/106
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 2048
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/106 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/30
--------------------------------------------------------------
64/107
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4*a6,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 2048
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/107 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/29
--------------------------------------------------------------
64/108
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4,a3^2=a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 1024
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/108 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/109
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/109 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/110
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4,a3^2=1,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
8 elements of order 4
48 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/110 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/32
--------------------------------------------------------------
64/111
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4,a3^2=1,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
8 elements of order 4
48 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/1
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/111 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/112
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a4,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/3
Commutator subgroup generated by [ a4 ] , type 2/1
Automorphism Group of size 512
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/112 --- 2/1 --- 1/1
--------------------------------------------------------------
64/113
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a4,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/113 --- 4/2 --- 1/1
--------------------------------------------------------------
64/114
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a4,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 256
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/114 --- 4/2 --- 1/1
--------------------------------------------------------------
64/115
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
20 elements of order 4
32 elements of order 8
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/2
Commutator subgroup generated by [ a4 ] , type 2/1
Automorphism Group of size 256
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/115 --- 2/1 --- 1/1
group is isomorphic to 8/1 x 8/4
--------------------------------------------------------------
64/116
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
20 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 128
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/116 --- 4/2 --- 1/1
--------------------------------------------------------------
64/117
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
20 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 256
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/117 --- 4/2 --- 1/1
--------------------------------------------------------------
64/118
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
28 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/118 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 4/1 x 16/12
--------------------------------------------------------------
64/119
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 256
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/119 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 4/1 x 16/13
--------------------------------------------------------------
64/120
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=a5*a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/120 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 4/1 x 16/14
--------------------------------------------------------------
64/121
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a6,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/121 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/122
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a6,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/122 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/123
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a5*a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a6,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
28 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/123 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/124
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a5,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
20 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a5, a6 ] , type 8/1
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 128
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/124 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/125
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a5,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a6,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
20 elements of order 4
32 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/1
Commutator subgroup generated by [ a4, a6 ] , type 4/1
Automorphism Group of size 128
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/125 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/126
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4,a3^2=a4*a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
28 elements of order 4
32 elements of order 8
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/2
Commutator subgroup generated by [ a4 ] , type 2/1
Automorphism Group of size 768
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/126 --- 2/1 --- 1/1
group is isomorphic to 8/1 x 8/5
--------------------------------------------------------------
64/127
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a4,a3^2=a4*a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
28 elements of order 4
32 elements of order 8
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a4, a6 ] , type 4/2
Automorphism Group of size 256
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/127 --- 4/2 --- 1/1
--------------------------------------------------------------
64/128
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
16 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/128 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/129
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a6,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/129 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/130
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
24 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/130 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/131
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
24 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/131 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/132
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a6,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/132 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/133
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/133 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/134
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
27 elements of order 2
20 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 32/33
Nilpotency class 3
Lower Central Series: 64/134 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/135
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
28 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 32/33
Nilpotency class 3
Lower Central Series: 64/135 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/136
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
28 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 32/33
Nilpotency class 3
Lower Central Series: 64/136 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/137
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 32/33
Nilpotency class 3
Lower Central Series: 64/137 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/138
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
27 elements of order 2
36 elements of order 4
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 384
Inner automorphism group type 32/33
Nilpotency class 3
Lower Central Series: 64/138 --- 8/3 --- 2/1 --- 1/1
--------------------------------------------------------------
64/139
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
44 elements of order 4
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/3
Automorphism Group of size 384
Inner automorphism group type 32/33
Nilpotency class 3
Lower Central Series: 64/139 --- 8/3 --- 2/1 --- 1/1
--------------------------------------------------------------
64/140
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
27 elements of order 2
20 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/140 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/141
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4,a3^2=a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
28 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/141 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/142
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/142 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/143
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4,a3^2=a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/143 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/144
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4*a6,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
28 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/144 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/145
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4*a6,a3^2=a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/145 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/146
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/146 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/147
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4*a6,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
24 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/147 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/148
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4*a6,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/148 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/149
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/149 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/150
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4*a6,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
24 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/150 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/151
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4*a6,a3^2=1,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/151 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/152
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4,a3^2=1,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5*a6,[a1,a4]=a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
16 elements of order 4
32 elements of order 8
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 128
Inner automorphism group type 32/36
Nilpotency class 3
Lower Central Series: 64/152 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/153
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4*a6,a3^2=1,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5*a6,[a1,a4]=a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
8 elements of order 4
32 elements of order 8
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 32/36
Nilpotency class 3
Lower Central Series: 64/153 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/154
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4*a6,a3^2=1,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5*a6,[a1,a4]=a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
24 elements of order 4
32 elements of order 8
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 32/36
Nilpotency class 3
Lower Central Series: 64/154 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/155
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/155 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/156
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4,a3^2=a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/156 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/157
((a1,a2,a3,a4,a5,a6;a1^2=a4*a6,a2^2=a4,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/157 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/158
((a1,a2,a3,a4,a5,a6;a1^2=a4*a6,a2^2=a4,a3^2=a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/158 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/159
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4*a6,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/159 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/160
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4*a6,a3^2=a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/160 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/161
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a4,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/161 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/162
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a4*a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/162 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/163
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a6,a3^2=a4,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/163 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/164
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=a4,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/164 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/165
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=a4*a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/165 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/166
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a6,a3^2=a4,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=a6,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
19 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 256
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/166 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/167
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5,a3^2=a4,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5*a6,[a1,a4]=a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
28 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a4*a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/167 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/168
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a5,a3^2=a4,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5*a6,[a1,a4]=a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a4*a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/168 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/169
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5*a6,a3^2=a4,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5*a6,[a1,a4]=a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a4*a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/169 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/170
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5,a3^2=a4,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5*a6,[a1,a4]=a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a4*a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/170 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/171
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5*a6,a3^2=a4,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5*a6,[a1,a4]=a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
28 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a4*a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/171 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/172
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a5*a6,a3^2=a4,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5*a6,[a1,a4]=a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a4*a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/172 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/173
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
28 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 2048
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/173 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/174
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4*a6,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
35 elements of order 2
12 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 4096
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/174 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/175
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4*a6,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 4096
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/175 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/176
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4,a3^2=a5*a6,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
28 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/176 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/177
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a4,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
27 elements of order 2
20 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/177 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/178
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a4,a3^2=a5,a4^2=a6,a5^2=1,a6^2=1
,[a1,a2]=a4*a6,[a1,a3]=a5,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
36 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/178 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/179
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4,a3^2=a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 2048
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/179 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/180
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4*a6,a3^2=a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/180 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/181
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4,a3^2=a5*a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 2048
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/181 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/182
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=a4,a3^2=a5,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a4,[a1,a3]=a5*a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
44 elements of order 4
16 elements of order 8
16 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/2
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/182 --- 8/2 --- 2/1 --- 1/1
--------------------------------------------------------------
64/183
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=1,a4^2=a5,a5^2=a6,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
8 elements of order 4
16 elements of order 8
32 elements of order 16
ABELIAN
group is isomorphic to 2/1 x 2/1 x 16/1
--------------------------------------------------------------
64/184
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=1,a4^2=a5,a5^2=a6,a6^2=1
,[a1,a2]=a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
8 elements of order 4
16 elements of order 8
32 elements of order 16
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/2
Commutator subgroup generated by [ a6 ] , type 2/1
Automorphism Group of size 256
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/184 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/22
--------------------------------------------------------------
64/185
((a1,a2,a3,a4,a5,a6;a1^2=a4,a2^2=1,a3^2=1,a4^2=a5,a5^2=a6,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
8 elements of order 4
16 elements of order 8
32 elements of order 16
40 conjugacy classes
Centre generated by [ a1, a4, a5, a6 ] , type 16/1
Commutator subgroup generated by [ a6 ] , type 2/1
Automorphism Group of size 192
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/185 --- 2/1 --- 1/1
--------------------------------------------------------------
64/186
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=a5*a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a5,[a1,a3]=1,[a1,a4]=a5*a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5*a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
35 elements of order 2
4 elements of order 4
8 elements of order 8
16 elements of order 16
22 conjugacy classes
Centre generated by [ a3, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/1
Automorphism Group of size 1024
Inner automorphism group type 16/12
Nilpotency class 4
Lower Central Series: 64/186 --- 8/1 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/49
--------------------------------------------------------------
64/187
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=1,a4^2=a5*a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a5,[a1,a3]=1,[a1,a4]=a5*a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5*a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
20 elements of order 4
8 elements of order 8
16 elements of order 16
22 conjugacy classes
Centre generated by [ a3, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/1
Automorphism Group of size 512
Inner automorphism group type 16/12
Nilpotency class 4
Lower Central Series: 64/187 --- 8/1 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/50
--------------------------------------------------------------
64/188
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a6,a3^2=1,a4^2=a5*a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a5,[a1,a3]=1,[a1,a4]=a5*a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5*a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
36 elements of order 4
8 elements of order 8
16 elements of order 16
22 conjugacy classes
Centre generated by [ a3, a6 ] , type 4/2
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/1
Automorphism Group of size 1024
Inner automorphism group type 16/12
Nilpotency class 4
Lower Central Series: 64/188 --- 8/1 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/51
--------------------------------------------------------------
64/189
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a6,a4^2=a5*a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a5,[a1,a3]=1,[a1,a4]=a5*a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5*a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
20 elements of order 4
8 elements of order 8
16 elements of order 16
22 conjugacy classes
Centre generated by [ a3, a6 ] , type 4/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/1
Automorphism Group of size 256
Inner automorphism group type 16/12
Nilpotency class 4
Lower Central Series: 64/189 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/190
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=a5*a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a5,[a1,a3]=a6,[a1,a4]=a5*a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5*a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
27 elements of order 2
12 elements of order 4
8 elements of order 8
16 elements of order 16
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/1
Automorphism Group of size 256
Inner automorphism group type 32/23
Nilpotency class 4
Lower Central Series: 64/190 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/191
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a6,a3^2=1,a4^2=a5*a6,a5^2=a6,a6^2=1
,[a1,a2]=a4*a5,[a1,a3]=a6,[a1,a4]=a5*a6,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5*a6,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
28 elements of order 4
8 elements of order 8
16 elements of order 16
16 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a4, a5, a6 ] , type 8/1
Automorphism Group of size 256
Inner automorphism group type 32/23
Nilpotency class 4
Lower Central Series: 64/191 --- 8/1 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/192
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
ABELIAN
group is isomorphic to 2/1 x 2/1 x 4/1 x 4/1
--------------------------------------------------------------
64/193
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
32 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/5
Commutator subgroup generated by [ a5 ] , type 2/1
Automorphism Group of size 49152
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/193 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 2/1 x 16/9
--------------------------------------------------------------
64/194
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a5,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/5
Commutator subgroup generated by [ a5 ] , type 2/1
Automorphism Group of size 49152
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/194 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 2/1 x 16/10
--------------------------------------------------------------
64/195
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a5 ] , type 2/1
Automorphism Group of size 8192
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/195 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/16
--------------------------------------------------------------
64/196
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a5 ] , type 2/1
Automorphism Group of size 4096
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/196 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 4/1 x 8/4
--------------------------------------------------------------
64/197
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a5,a3^2=a5*a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a5 ] , type 2/1
Automorphism Group of size 12288
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/197 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 4/1 x 8/5
--------------------------------------------------------------
64/198
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=a6,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/3
Commutator subgroup generated by [ a5 ] , type 2/1
Automorphism Group of size 3072
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/198 --- 2/1 --- 1/1
group is isomorphic to 4/1 x 16/8
--------------------------------------------------------------
64/199
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=1,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
34 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5 ] , type 2/1
Automorphism Group of size 4096
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/199 --- 2/1 --- 1/1
--------------------------------------------------------------
64/200
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a5,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=1,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
34 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5 ] , type 2/1
Automorphism Group of size 12288
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/200 --- 2/1 --- 1/1
--------------------------------------------------------------
64/201
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=1,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=1,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
34 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5 ] , type 2/1
Automorphism Group of size 3072
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/201 --- 2/1 --- 1/1
--------------------------------------------------------------
64/202
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
39 elements of order 2
24 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 12288
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/202 --- 4/2 --- 1/1
group is isomorphic to 2/1 x 32/33
--------------------------------------------------------------
64/203
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
32 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 4096
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/203 --- 4/2 --- 1/1
group is isomorphic to 2/1 x 32/36
--------------------------------------------------------------
64/204
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a5,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 4096
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/204 --- 4/2 --- 1/1
group is isomorphic to 2/1 x 32/37
--------------------------------------------------------------
64/205
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 4096
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/205 --- 4/2 --- 1/1
group is isomorphic to 2/1 x 32/38
--------------------------------------------------------------
64/206
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/206 --- 4/2 --- 1/1
--------------------------------------------------------------
64/207
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a6,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 8192
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/207 --- 4/2 --- 1/1
group is isomorphic to 2/1 x 32/39
--------------------------------------------------------------
64/208
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 8192
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/208 --- 4/2 --- 1/1
group is isomorphic to 2/1 x 32/40
--------------------------------------------------------------
64/209
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5*a6,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 6144
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/209 --- 4/2 --- 1/1
group is isomorphic to 2/1 x 32/41
--------------------------------------------------------------
64/210
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a6,a3^2=1,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/210 --- 4/2 --- 1/1
--------------------------------------------------------------
64/211
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
39 elements of order 2
24 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 49152
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/211 --- 4/2 --- 1/1
group is isomorphic to 2/1 x 32/34
--------------------------------------------------------------
64/212
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a5,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 16384
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/212 --- 4/2 --- 1/1
group is isomorphic to 2/1 x 32/35
--------------------------------------------------------------
64/213
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a6,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/213 --- 4/2 --- 1/1
--------------------------------------------------------------
64/214
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=a6,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
28 conjugacy classes
Centre generated by [ a4, a5, a6 ] , type 8/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 8/3
Nilpotency class 2
Lower Central Series: 64/214 --- 4/2 --- 1/1
--------------------------------------------------------------
64/215
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
32 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/215 --- 4/2 --- 1/1
--------------------------------------------------------------
64/216
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
32 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/216 --- 4/2 --- 1/1
--------------------------------------------------------------
64/217
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/217 --- 4/2 --- 1/1
--------------------------------------------------------------
64/218
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/218 --- 4/2 --- 1/1
--------------------------------------------------------------
64/219
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a6,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/219 --- 4/2 --- 1/1
--------------------------------------------------------------
64/220
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a6,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/220 --- 4/2 --- 1/1
--------------------------------------------------------------
64/221
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a6,a3^2=1,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/221 --- 4/2 --- 1/1
--------------------------------------------------------------
64/222
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=1,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/222 --- 4/2 --- 1/1
--------------------------------------------------------------
64/223
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5*a6,a3^2=1,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/223 --- 4/2 --- 1/1
--------------------------------------------------------------
64/224
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 6144
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/224 --- 4/2 --- 1/1
--------------------------------------------------------------
64/225
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a5,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
56 elements of order 4
22 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/225 --- 4/2 --- 1/1
--------------------------------------------------------------
64/226
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
35 elements of order 2
28 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/226 --- 4/2 --- 1/1
group is isomorphic to 8/4 x 8/4
--------------------------------------------------------------
64/227
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
27 elements of order 2
36 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/227 --- 4/2 --- 1/1
--------------------------------------------------------------
64/228
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
44 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/228 --- 4/2 --- 1/1
--------------------------------------------------------------
64/229
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5,a3^2=1,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
44 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 1536
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/229 --- 4/2 --- 1/1
--------------------------------------------------------------
64/230
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a5,a3^2=1,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
52 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 3072
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/230 --- 4/2 --- 1/1
group is isomorphic to 8/4 x 8/5
--------------------------------------------------------------
64/231
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5,a3^2=a5,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
27 elements of order 2
36 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 3072
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/231 --- 4/2 --- 1/1
--------------------------------------------------------------
64/232
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
44 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/232 --- 4/2 --- 1/1
--------------------------------------------------------------
64/233
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
52 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/233 --- 4/2 --- 1/1
--------------------------------------------------------------
64/234
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5*a6,a3^2=1,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
44 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 512
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/234 --- 4/2 --- 1/1
--------------------------------------------------------------
64/235
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a5*a6,a3^2=1,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
52 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/235 --- 4/2 --- 1/1
--------------------------------------------------------------
64/236
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a6,a3^2=a5,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
44 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/236 --- 4/2 --- 1/1
--------------------------------------------------------------
64/237
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a5*a6,a3^2=a5,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
52 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/237 --- 4/2 --- 1/1
--------------------------------------------------------------
64/238
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a5,a3^2=a6,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
60 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 3072
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/238 --- 4/2 --- 1/1
--------------------------------------------------------------
64/239
((a1,a2,a3,a4,a5,a6;a1^2=a5*a6,a2^2=a5,a3^2=a6,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
60 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 18432
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/239 --- 4/2 --- 1/1
group is isomorphic to 8/5 x 8/5
--------------------------------------------------------------
64/240
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a5*a6,a3^2=a6,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
44 elements of order 4
25 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 2048
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/240 --- 4/2 --- 1/1
--------------------------------------------------------------
64/241
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5*a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5*a6,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
27 elements of order 2
36 elements of order 4
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 1536
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/241 --- 4/2 --- 1/1
--------------------------------------------------------------
64/242
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a5,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5*a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5*a6,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
27 elements of order 2
36 elements of order 4
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 9216
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/242 --- 4/2 --- 1/1
--------------------------------------------------------------
64/243
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5*a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5*a6,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
19 elements of order 2
44 elements of order 4
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/243 --- 4/2 --- 1/1
--------------------------------------------------------------
64/244
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a5,a3^2=a5,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5*a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5*a6,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
11 elements of order 2
52 elements of order 4
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 1024
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/244 --- 4/2 --- 1/1
--------------------------------------------------------------
64/245
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=a6,a3^2=a6,a4^2=a5,a5^2=1,a6^2=1
,[a1,a2]=a5,[a1,a3]=a6,[a1,a4]=a5*a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a5*a6,[a2,a4]=a5,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
3 elements of order 2
60 elements of order 4
19 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/2
Automorphism Group of size 15360
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/245 --- 4/2 --- 1/1
--------------------------------------------------------------
64/246
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
16 elements of order 4
32 elements of order 8
ABELIAN
group is isomorphic to 2/1 x 2/1 x 2/1 x 8/1
--------------------------------------------------------------
64/247
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
16 elements of order 4
32 elements of order 8
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a6 ] , type 2/1
Automorphism Group of size 6144
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/247 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 2/1 x 16/11
--------------------------------------------------------------
64/248
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
16 elements of order 4
32 elements of order 8
40 conjugacy classes
Centre generated by [ a1, a4, a5, a6 ] , type 16/2
Commutator subgroup generated by [ a6 ] , type 2/1
Automorphism Group of size 1536
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/248 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/17
--------------------------------------------------------------
64/249
((a1,a2,a3,a4,a5,a6;a1^2=a5,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
16 elements of order 4
32 elements of order 8
34 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/1
Commutator subgroup generated by [ a6 ] , type 2/1
Automorphism Group of size 1536
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/249 --- 2/1 --- 1/1
--------------------------------------------------------------
64/250
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a5*a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
39 elements of order 2
8 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a4, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/1
Automorphism Group of size 12288
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/250 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 2/1 x 16/12
--------------------------------------------------------------
64/251
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a5*a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
24 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a4, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/1
Automorphism Group of size 6144
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/251 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 2/1 x 16/13
--------------------------------------------------------------
64/252
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a6,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a5*a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
7 elements of order 2
40 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a4, a6 ] , type 8/3
Commutator subgroup generated by [ a5, a6 ] , type 4/1
Automorphism Group of size 12288
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/252 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 2/1 x 16/14
--------------------------------------------------------------
64/253
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a6,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a5*a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
24 elements of order 4
16 elements of order 8
28 conjugacy classes
Centre generated by [ a3, a4, a6 ] , type 8/2
Commutator subgroup generated by [ a5, a6 ] , type 4/1
Automorphism Group of size 1024
Inner automorphism group type 8/4
Nilpotency class 3
Lower Central Series: 64/253 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/26
--------------------------------------------------------------
64/254
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a5*a6,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
16 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/1
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/254 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/44
--------------------------------------------------------------
64/255
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a6,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a5*a6,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/2
Commutator subgroup generated by [ a5, a6 ] , type 4/1
Automorphism Group of size 1024
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/255 --- 4/1 --- 2/1 --- 1/1
group is isomorphic to 2/1 x 32/45
--------------------------------------------------------------
64/256
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=a6,a5^2=a6,a6^2=1
,[a1,a2]=a5*a6,[a1,a3]=a6,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
24 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a4, a6 ] , type 4/1
Commutator subgroup generated by [ a5, a6 ] , type 4/1
Automorphism Group of size 512
Inner automorphism group type 16/6
Nilpotency class 3
Lower Central Series: 64/256 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/257
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a5*a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
16 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a5, a6 ] , type 4/1
Automorphism Group of size 768
Inner automorphism group type 32/8
Nilpotency class 3
Lower Central Series: 64/257 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/258
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a5*a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
24 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a5, a6 ] , type 4/1
Automorphism Group of size 384
Inner automorphism group type 32/8
Nilpotency class 3
Lower Central Series: 64/258 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/259
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a6,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a5*a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=a6,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=a6,[a2,a6]=1
,[a3,a4]=a6,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
32 elements of order 4
16 elements of order 8
22 conjugacy classes
Centre generated by [ a6 ] , type 2/1
Commutator subgroup generated by [ a5, a6 ] , type 4/1
Automorphism Group of size 768
Inner automorphism group type 32/8
Nilpotency class 3
Lower Central Series: 64/259 --- 4/1 --- 2/1 --- 1/1
--------------------------------------------------------------
64/260
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
32 elements of order 4
ABELIAN
group is isomorphic to 2/1 x 2/1 x 2/1 x 2/1 x 4/1
--------------------------------------------------------------
64/261
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
47 elements of order 2
16 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/5
Commutator subgroup generated by [ a6 ] , type 2/1
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/261 --- 2/1 --- 1/1
--------------------------------------------------------------
64/262
((a1,a2,a3,a4,a5,a6;a1^2=a6,a2^2=a6,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
15 elements of order 2
48 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/5
Commutator subgroup generated by [ a6 ] , type 2/1
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/262 --- 2/1 --- 1/1
--------------------------------------------------------------
64/263
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a6,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
32 elements of order 4
40 conjugacy classes
Centre generated by [ a3, a4, a5, a6 ] , type 16/4
Commutator subgroup generated by [ a6 ] , type 2/1
Inner automorphism group type 4/2
Nilpotency class 2
Lower Central Series: 64/263 --- 2/1 --- 1/1
--------------------------------------------------------------
64/264
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
39 elements of order 2
24 elements of order 4
34 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a6 ] , type 2/1
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/264 --- 2/1 --- 1/1
--------------------------------------------------------------
64/265
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=a6,a3^2=a6,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
23 elements of order 2
40 elements of order 4
34 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/2
Commutator subgroup generated by [ a6 ] , type 2/1
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/265 --- 2/1 --- 1/1
--------------------------------------------------------------
64/266
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=a6,a6^2=1
,[a1,a2]=a6,[a1,a3]=1,[a1,a4]=a6,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=a6,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
31 elements of order 2
32 elements of order 4
34 conjugacy classes
Centre generated by [ a5, a6 ] , type 4/1
Commutator subgroup generated by [ a6 ] , type 2/1
Inner automorphism group type 16/5
Nilpotency class 2
Lower Central Series: 64/266 --- 2/1 --- 1/1
--------------------------------------------------------------
64/267
((a1,a2,a3,a4,a5,a6;a1^2=1,a2^2=1,a3^2=1,a4^2=1,a5^2=1,a6^2=1
,[a1,a2]=1,[a1,a3]=1,[a1,a4]=1,[a1,a5]=1,[a1,a6]=1
,[a2,a3]=1,[a2,a4]=1,[a2,a5]=1,[a2,a6]=1
,[a3,a4]=1,[a3,a5]=1,[a3,a6]=1
,[a4,a5]=1,[a4,a6]=1
,[a5,a6]=1
))
63 elements of order 2
ABELIAN