Small group number 8935 of order 256
G is the group 256gp8935
G has 4 minimal generators, rank 6 and exponent 4.
The centre has rank 4.
The 15 maximal subgroups are:
128gp170 (9x), 128gp2163 (6x).
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
6, 6.
This cohomology ring calculation is only complete
out to degree 4.
Ring structure
| Restriction information
The cohomology ring has 12 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- y4 in degree 1
- x1 in degree 2
- x2 in degree 2
- x3 in degree 2
- x4 in degree 2
- x5 in degree 2, a regular element
- x6 in degree 2, a regular element
- x7 in degree 2, a regular element
- x8 in degree 2, a regular element
There are 19 minimal relations:
- y42 =
y3.y4
- y2.y4 =
0
- y1.y4 =
y1.y3
- y1.y2 =
0
- y4.x4 =
y3.x4
+ y1.x2
- y4.x3 =
y3.x3
- y4.x2 =
0
- y4.x1 =
y1.x2
- y2.x4 =
0
- y2.x3 =
y1.x2
- y1.x1 =
0
- x42 =
y1.y3.x4
+ y3.y4.x7
+ y12.x5
- x32 =
y1.y3.x3
+ y3.y4.x8
+ y12.x6
- x2.x4 =
0
- x2.x3 =
0
- x22 =
y2.y3.x2
+ y3.y4.x5
+ y32.x5
+ y22.x6
- x1.x4 =
0
- x1.x3 =
0
- x12 =
y2.y3.x1
+ y3.y4.x7
+ y32.x7
+ y22.x8
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
Essential ideal:
Zero ideal
Nilradical:
There is one minimal generator:
Restriction to maximal subgroup number 1, which is 128gp2163
- y1 restricts to
y3
- y2 restricts to
y1
- y3 restricts to
y2
- y4 restricts to
0
- x1 restricts to
x1
+ y22
+ y12
+ y2.y4
+ y1.y4
- x2 restricts to
y22
+ y12
+ y2.y4
+ y1.y5
+ y1.y4
- x3 restricts to
y3.y5
+ y3.y4
- x4 restricts to
y3.y4
- x5 restricts to
y22
+ y1.y2
+ y1.y4
+ y42
- x6 restricts to
y1.y2
+ y12
+ y2.y5
+ y2.y4
+ y52
+ y42
- x7 restricts to
y22
+ y1.y2
+ x3
+ y3.y4
+ y1.y4
+ y42
- x8 restricts to
y1.y2
+ y12
+ x2
+ y3.y4
+ y2.y4
+ y42
Restriction to maximal subgroup number 2, which is 128gp2163
- y1 restricts to
0
- y2 restricts to
y1
- y3 restricts to
y3
+ y2
- y4 restricts to
y3
- x1 restricts to
y22
+ y1.y2
+ y2.y4
+ y1.y5
- x2 restricts to
x1
+ y22
+ y1.y2
+ y12
+ y2.y4
+ y1.y4
- x3 restricts to
y3.y5
- x4 restricts to
y32
+ y3.y4
- x5 restricts to
y22
+ y1.y2
+ x3
+ y3.y4
+ y1.y4
+ y42
- x6 restricts to
y1.y2
+ y12
+ x2
+ y3.y4
+ y2.y4
+ y42
- x7 restricts to
y32
+ y22
+ y1.y2
+ y1.y4
+ y42
- x8 restricts to
y2.y5
+ y52
Restriction to maximal subgroup number 3, which is 128gp2163
- y1 restricts to
y3
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
+ y1
- y4 restricts to
y3
- x1 restricts to
x1
+ y22
+ y1.y2
- x2 restricts to
x1
+ y22
+ y12
+ y2.y5
+ y2.y4
+ y1.y4
- x3 restricts to
y3.y5
+ y3.y4
- x4 restricts to
y3.y5
- x5 restricts to
x1
+ x3
+ x2
+ y3.y5
+ y2.y5
+ y1.y5
+ y52
- x6 restricts to
y22
+ y1.y2
+ x3
+ y3.y5
+ y3.y4
+ y1.y5
+ y1.y4
+ y52
+ y42
- x7 restricts to
x1
+ x3
+ x2
- x8 restricts to
y22
+ y1.y2
+ x3
Restriction to maximal subgroup number 4, which is 128gp2163
- y1 restricts to
y2
+ y1
- y2 restricts to
0
- y3 restricts to
y3
+ y1
- y4 restricts to
y1
- x1 restricts to
y32
+ y3.y5
- x2 restricts to
y3.y4
- x3 restricts to
x1
+ y1.y2
+ y12
+ y1.y5
+ y1.y4
- x4 restricts to
y22
+ y12
+ y2.y4
+ y1.y5
+ y1.y4
- x5 restricts to
y22
+ y1.y2
+ y1.y4
+ y42
- x6 restricts to
x3
- x7 restricts to
y32
+ y2.y5
+ y1.y5
+ y52
- x8 restricts to
x1
+ x3
+ x2
+ y3.y5
+ y3.y4
+ y2.y5
+ y2.y4
+ y1.y5
+ y1.y4
+ y52
+ y42
Restriction to maximal subgroup number 5, which is 128gp170
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y3
- y4 restricts to
y1
- x1 restricts to
x2
+ x1
- x2 restricts to
x4
- x3 restricts to
x3
+ x2
+ y22
- x4 restricts to
x3
- x5 restricts to
x5
- x6 restricts to
x4
+ y22
+ x6
+ x5
- x7 restricts to
x7
- x8 restricts to
x2
+ x1
+ x8
+ x7
Restriction to maximal subgroup number 6, which is 128gp170
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y3
+ y2
+ y1
- y4 restricts to
y2
+ y1
- x1 restricts to
x4
- x2 restricts to
x4
+ x2
+ x1
- x3 restricts to
x2
- x4 restricts to
x3
+ y22
- x5 restricts to
x3
+ x7
+ x5
- x6 restricts to
x2
+ x8
+ x6
- x7 restricts to
y22
+ x5
- x8 restricts to
x6
Restriction to maximal subgroup number 7, which is 128gp170
- y1 restricts to
y2
- y2 restricts to
y1
- y3 restricts to
y3
+ y2
- y4 restricts to
y2
+ y1
- x1 restricts to
x2
+ x1
- x2 restricts to
x4
+ x2
+ x1
- x3 restricts to
x3
+ x2
- x4 restricts to
x3
+ y22
- x5 restricts to
x3
+ x7
+ x5
- x6 restricts to
x4
+ x3
+ x1
+ x8
+ x7
+ x6
+ x5
- x7 restricts to
x7
- x8 restricts to
x2
+ x1
+ x8
+ x7
Restriction to maximal subgroup number 8, which is 128gp170
- y1 restricts to
y2
+ y1
- y2 restricts to
y3
+ y1
- y3 restricts to
0
- y4 restricts to
y1
- x1 restricts to
x4
+ x2
+ x1
- x2 restricts to
x4
+ y32
- x3 restricts to
x3
+ y22
- x4 restricts to
x3
+ x2
+ y22
- x5 restricts to
x4
+ y22
+ x6
+ x5
- x6 restricts to
y32
+ y22
+ x5
- x7 restricts to
x4
+ x3
+ x1
+ x8
+ x7
+ x6
+ x5
- x8 restricts to
x3
+ x7
+ x5
Restriction to maximal subgroup number 9, which is 128gp2163
- y1 restricts to
y1
- y2 restricts to
y3
- y3 restricts to
y2
- y4 restricts to
y2
- x1 restricts to
y3.y5
- x2 restricts to
y3.y4
- x3 restricts to
y1.y2
+ y12
+ y2.y5
+ y1.y4
- x4 restricts to
x1
+ y1.y2
+ y2.y5
+ y1.y4
- x5 restricts to
x2
+ y3.y4
+ y2.y4
+ y42
- x6 restricts to
y1.y2
+ y12
+ y2.y4
+ y42
- x7 restricts to
x3
+ y3.y5
+ y1.y5
+ y52
- x8 restricts to
y1.y5
+ y52
Restriction to maximal subgroup number 10, which is 128gp170
- y1 restricts to
y2
- y2 restricts to
y3
+ y1
- y3 restricts to
y2
- y4 restricts to
y2
+ y1
- x1 restricts to
x2
+ x1
+ y32
- x2 restricts to
x4
+ x2
+ x1
- x3 restricts to
x3
- x4 restricts to
x3
+ x2
+ y22
- x5 restricts to
x4
+ x3
+ x1
+ x8
+ x7
+ x6
+ x5
- x6 restricts to
x3
+ x7
+ x5
- x7 restricts to
x2
+ x1
+ x8
+ x7
- x8 restricts to
y32
+ x7
Restriction to maximal subgroup number 11, which is 128gp170
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y3
+ y1
- x1 restricts to
x3
+ y22
- x2 restricts to
x3
+ x2
- x3 restricts to
x4
- x4 restricts to
x4
+ x2
+ x1
- x5 restricts to
x4
+ x3
+ x1
+ x8
+ x7
+ x6
+ x5
- x6 restricts to
x4
+ x6
+ x5
- x7 restricts to
x3
+ x7
+ x5
- x8 restricts to
y22
+ x5
Restriction to maximal subgroup number 12, which is 128gp170
- y1 restricts to
y2
+ y1
- y2 restricts to
y3
+ y1
- y3 restricts to
y3
+ y1
- y4 restricts to
y1
- x1 restricts to
x4
+ x2
+ x1
- x2 restricts to
x4
- x3 restricts to
x2
- x4 restricts to
x3
+ x2
- x5 restricts to
x4
+ x6
+ x5
- x6 restricts to
x6
- x7 restricts to
x4
+ x3
+ x1
+ x8
+ x7
+ x6
+ x5
- x8 restricts to
x2
+ x8
+ x6
Restriction to maximal subgroup number 13, which is 128gp2163
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
y3
+ y1
- y4 restricts to
y1
- x1 restricts to
y32
+ y3.y4
- x2 restricts to
y32
+ y3.y5
- x3 restricts to
x1
+ y22
+ y1.y2
+ y12
+ y2.y5
+ y2.y4
- x4 restricts to
x1
+ y12
+ y2.y4
+ y1.y4
- x5 restricts to
x3
+ y3.y4
+ y1.y4
+ y42
- x6 restricts to
y22
+ y1.y2
+ x3
+ y3.y5
+ y3.y4
+ y1.y5
+ y1.y4
+ y52
+ y42
- x7 restricts to
y1.y2
+ y12
+ x2
+ y3.y4
+ y2.y4
+ y42
- x8 restricts to
y1.y2
+ y12
+ x2
Restriction to maximal subgroup number 14, which is 128gp170
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
y3
+ y2
- y4 restricts to
y2
+ y1
- x1 restricts to
x2
+ x1
+ y32
- x2 restricts to
x4
+ x2
+ x1
- x3 restricts to
x3
+ x2
- x4 restricts to
x2
+ y22
- x5 restricts to
x2
+ x8
+ x6
- x6 restricts to
x4
+ x3
+ x1
+ x8
+ x7
+ x6
+ x5
- x7 restricts to
x8
- x8 restricts to
x2
+ x1
+ x8
+ x7
Restriction to maximal subgroup number 15, which is 128gp170
- y1 restricts to
y1
- y2 restricts to
y3
+ y1
- y3 restricts to
y3
+ y2
- y4 restricts to
y2
- x1 restricts to
x4
+ y32
- x2 restricts to
x2
+ x1
- x3 restricts to
x2
+ y22
- x4 restricts to
x3
+ x2
+ y22
- x5 restricts to
x2
+ x1
+ x8
+ x7
- x6 restricts to
x8
- x7 restricts to
x4
+ y22
+ x6
+ x5
- x8 restricts to
y22
+ x6
Restriction to maximal elementary abelian number 1, which is V64
- y1 restricts to
y5
- y2 restricts to
0
- y3 restricts to
y6
- y4 restricts to
y6
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
y5.y6
+ y4.y5
+ y1.y6
- x4 restricts to
y3.y6
+ y2.y5
- x5 restricts to
y2.y6
+ y22
- x6 restricts to
y4.y6
+ y42
- x7 restricts to
y3.y5
+ y32
- x8 restricts to
y1.y5
+ y12
Restriction to maximal elementary abelian number 2, which is V64
- y1 restricts to
0
- y2 restricts to
y5
- y3 restricts to
y6
- y4 restricts to
0
- x1 restricts to
y3.y6
+ y1.y5
- x2 restricts to
y4.y5
+ y2.y6
- x3 restricts to
0
- x4 restricts to
0
- x5 restricts to
y2.y5
+ y22
- x6 restricts to
y4.y6
+ y42
- x7 restricts to
y3.y5
+ y32
- x8 restricts to
y1.y6
+ y12
Restriction to the greatest central elementary abelian, which is V16
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
0
- x5 restricts to
y22
- x6 restricts to
y42
- x7 restricts to
y32
- x8 restricts to
y12
Back to the groups of order 256