Small group number 256 of order 64
G is the group 64gp256
The Hall-Senior number of this group is 112.
G has 4 minimal generators, rank 3 and exponent 8.
The centre has rank 1.
The 15 maximal subgroups are:
32gp37, 32gp42 (4x), 32gp43 (4x), 32gp44 (4x), 32gp48 (2x).
There are 3 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 7 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- y4 in degree 1
- u1 in degree 5
- u2 in degree 5
- r in degree 8, a regular element
There are 9 minimal relations:
- y1.y4 =
0
- y32.y4 =
y22.y4
+ y1.y32
- y1.y22.y32 =
0
- y4.u1 =
0
- y1.u2 =
0
- y32.u2 =
y32.u1
+ y22.u2
- u22 =
y24.y4.u2
+ y24.y36
+ y26.y44
+ y29.y4
+ y42.r
- u1.u2 =
y24.y36
+ y26.y34
- u12 =
y24.y36
+ y28.y32
+ y1.y24.u1
+ y12.y23.u1
+ y14.y2.u1
+ y12.r
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y12.y32 =
0
- y1.y32.u1 =
0
Essential ideal:
Zero ideal
Nilradical:
There are 4 minimal generators:
-
y3.y4
+ y2.y4
-
y1.y3
-
y3.u2
+ y2.u2
+ y22.y34
+ y23.y33
-
y3.u1
+ y22.y34
+ y24.y32
This cohomology ring was obtained from a calculation
out to degree 13. The cohomology ring approximation
is stable from degree 10 onwards, and
Carlson's tests detect stability from degree 13
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
y42
+ y3.y4
+ y32
+ y2.y3
+ y22
+ y1.y2
+ y12
in degree 2
- h3 =
y2.y42
+ y2.y32
+ y22.y4
+ y22.y3
+ y1.y22
+ y12.y2
in degree 3
The first
2 terms h1, h2 form
a regular sequence of maximum length.
The remaining
term h3 is
annihilated by the class
y1.y2.y32.
The first
term h1 forms
a complete Duflot regular sequence.
That is, its restriction to the greatest central elementary abelian
subgroup forms a regular sequence of maximal length.
The ideal of essential classes is
the zero ideal.
The essential ideal squares to zero.
A basis for R/(h1, h2, h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 13.
-
1
in degree 0
-
y4
in degree 1
-
y3
in degree 1
-
y2
in degree 1
-
y1
in degree 1
-
y3.y4
in degree 2
-
y32
in degree 2
-
y2.y4
in degree 2
-
y2.y3
in degree 2
-
y22
in degree 2
-
y1.y3
in degree 2
-
y1.y2
in degree 2
-
y12
in degree 2
-
y33
in degree 3
-
y2.y32
in degree 3
-
y22.y4
in degree 3
-
y22.y3
in degree 3
-
y23
in degree 3
-
y1.y2.y3
in degree 3
-
y1.y22
in degree 3
-
y12.y3
in degree 3
-
y12.y2
in degree 3
-
y13
in degree 3
-
y22.y32
in degree 4
-
y23.y4
in degree 4
-
y24
in degree 4
-
y1.y22.y3
in degree 4
-
y12.y22
in degree 4
-
y13.y3
in degree 4
-
y13.y2
in degree 4
-
y14
in degree 4
-
u2
in degree 5
-
u1
in degree 5
-
y24.y4
in degree 5
-
y14.y2
in degree 5
-
y15
in degree 5
-
y4.u2
in degree 6
-
y3.u2
in degree 6
-
y3.u1
in degree 6
-
y2.u2
in degree 6
-
y2.u1
in degree 6
-
y1.u1
in degree 6
-
y3.y4.u2
in degree 7
-
y2.y4.u2
in degree 7
-
y2.y3.u2
in degree 7
-
y2.y3.u1
in degree 7
-
y22.u2
in degree 7
-
y22.u1
in degree 7
-
y1.y3.u1
in degree 7
-
y1.y2.u1
in degree 7
-
y12.u1
in degree 7
-
y22.y4.u2
in degree 8
-
y22.y3.u2
in degree 8
-
y22.y3.u1
in degree 8
-
y23.u2
in degree 8
-
y1.y22.u1
in degree 8
-
y12.y3.u1
in degree 8
-
y12.y2.u1
in degree 8
-
y13.u1
in degree 8
-
y24.u2
in degree 9
-
y12.y22.u1
in degree 9
-
y13.y2.u1
in degree 9
-
y14.u1
in degree 9
-
y14.y2.u1
in degree 10
A basis for AnnR/(h1, h2)(h3) is as follows.
Carlson's Koszul condition stipulates that this must
be confined to degrees less than 10.
-
y1.y2.y3
+ y1.y22
+ y12.y2
+ y13
in degree 3
-
y1.y22.y3
+ y1.y23
+ y12.y22
+ y13.y2
in degree 4
-
y1.y23
+ y13.y3
+ y14
in degree 4
-
y1.y24
+ y13.y2.y3
+ y14.y2
in degree 5
Restriction to maximal subgroup number 1, which is 32gp48
- y1 restricts to
y3
+ y2
- y2 restricts to
y3
+ y1
+ y4
- y3 restricts to
y1
- y4 restricts to
0
- u1 restricts to
y3.v
+ y2.v
+ y23.y42
+ y2.y44
- u2 restricts to
y12.y2.y42
- r restricts to
y34.v
+ y1.y23.v
+ y13.y2.v
+ y14.v
+ v2
+ y23.y4.v
+ y1.y25.y42
+ y12.y2.y4.v
+ y32.y42.v
+ y25.y43
+ y1.y3.y42.v
+ y1.y2.y42.v
+ y3.y43.v
+ y2.y43.v
+ y24.y44
+ y14.y44
+ y44.v
+ y22.y46
+ y1.y2.y46
+ y2.y47
Restriction to maximal subgroup number 2, which is 32gp48
- y1 restricts to
0
- y2 restricts to
y4
- y3 restricts to
y1
+ y4
- y4 restricts to
y3
+ y2
- u1 restricts to
y12.y2.y42
+ y12.y43
- u2 restricts to
y3.v
+ y2.v
+ y33.y42
+ y12.y43
+ y3.y44
+ y45
- r restricts to
v2
+ y16.y42
+ y12.y42.v
+ y14.y2.y43
+ y34.y44
+ y44.v
+ y22.y46
+ y12.y46
Restriction to maximal subgroup number 3, which is 32gp37
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
+ y1
- y4 restricts to
y1
- u1 restricts to
y24.y3
+ y22.y33
+ y22.w
+ y1.v
+ y32.w
+ y1.y2.y33
- u2 restricts to
y22.y33
+ y1.v
+ y32.w
+ y1.y2.y33
- r restricts to
y27.y3
+ v2
+ y22.y32.v
+ y24.y34
+ y34.v
+ y23.y35
+ y38
+ y25.w
+ y23.y32.w
Restriction to maximal subgroup number 4, which is 32gp42
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y2
- y4 restricts to
y3
- u1 restricts to
y1.v
- u2 restricts to
y3.v
- r restricts to
y28
+ v2
Restriction to maximal subgroup number 5, which is 32gp42
- y1 restricts to
y3
- y2 restricts to
y1
- y3 restricts to
y2
+ y1
- y4 restricts to
y1
- u1 restricts to
y3.v
- u2 restricts to
y15
+ y1.v
- r restricts to
y28
+ y18
+ y14.v
+ v2
Restriction to maximal subgroup number 6, which is 32gp42
- y1 restricts to
y1
- y2 restricts to
y1
- y3 restricts to
y2
- y4 restricts to
y3
- u1 restricts to
y1.v
- u2 restricts to
y3.v
- r restricts to
y28
+ y14.v
+ v2
Restriction to maximal subgroup number 7, which is 32gp42
- y1 restricts to
y3
- y2 restricts to
y3
+ y1
- y3 restricts to
y2
+ y1
- y4 restricts to
y1
- u1 restricts to
y3.v
- u2 restricts to
y15
+ y1.v
- r restricts to
y28
+ y18
+ y34.v
+ y14.v
+ v2
Restriction to maximal subgroup number 8, which is 32gp43
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
0
- y4 restricts to
y3
- u1 restricts to
y22.w
+ y1.v
- u2 restricts to
y3.v
- r restricts to
y25.w
+ y12.y23.w
+ y1.y23.v
+ y13.y2.v
+ v2
Restriction to maximal subgroup number 9, which is 32gp43
- y1 restricts to
y1
- y2 restricts to
y3
+ y2
- y3 restricts to
y3
- y4 restricts to
y3
- u1 restricts to
y22.w
+ y1.v
- u2 restricts to
y35
+ y2.y34
+ y3.v
- r restricts to
y38
+ y25.w
+ y12.y23.w
+ y34.v
+ y1.y23.v
+ y13.y2.v
+ v2
Restriction to maximal subgroup number 10, which is 32gp44
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y1
- y4 restricts to
y3
- u1 restricts to
u1
- u2 restricts to
u2
- r restricts to
r
+ y12.y2.u1
Restriction to maximal subgroup number 11, which is 32gp44
- y1 restricts to
y1
- y2 restricts to
y3
+ y2
+ y1
- y3 restricts to
y3
+ y1
- y4 restricts to
y3
- u1 restricts to
u1
+ y1.y24
- u2 restricts to
u2
+ y35
+ y2.y34
- r restricts to
y33.u2
+ y38
+ r
+ y1.y27
+ y12.y2.u1
Restriction to maximal subgroup number 12, which is 32gp43
- y1 restricts to
y3
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
+ y1
- y4 restricts to
y1
- u1 restricts to
y3.v
- u2 restricts to
y22.w
+ y25
+ y13.y22
+ y14.y2
+ y15
+ y1.v
- r restricts to
y13.y22.w
+ y14.y24
+ y18
+ y2.y33.v
+ y14.v
+ v2
Restriction to maximal subgroup number 13, which is 32gp44
- y1 restricts to
y3
- y2 restricts to
y2
- y3 restricts to
y2
+ y1
- y4 restricts to
y1
- u1 restricts to
u2
+ y12.y23
- u2 restricts to
y25
+ u1
+ y1.y24
+ y12.y23
- r restricts to
y2.y32.u2
+ r
+ y23.u1
+ y1.y22.u1
Restriction to maximal subgroup number 14, which is 32gp43
- y1 restricts to
y3
- y2 restricts to
y3
+ y2
+ y1
- y3 restricts to
y2
+ y1
- y4 restricts to
y1
- u1 restricts to
y3.v
- u2 restricts to
y22.w
+ y25
+ y13.y22
+ y14.y2
+ y15
+ y1.v
- r restricts to
y13.y22.w
+ y14.y24
+ y18
+ y34.v
+ y14.v
+ v2
Restriction to maximal subgroup number 15, which is 32gp44
- y1 restricts to
y3
- y2 restricts to
y3
+ y2
+ y1
- y3 restricts to
y2
- y4 restricts to
y1
- u1 restricts to
u2
+ y12.y23
- u2 restricts to
y25
+ u1
+ y1.y24
+ y12.y23
- r restricts to
y33.u2
+ r
+ y23.u1
+ y1.y22.u1
+ y12.y26
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
y3
- y4 restricts to
y2
- u1 restricts to
0
- u2 restricts to
y35
+ y12.y23
+ y14.y2
- r restricts to
y22.y36
+ y12.y22.y34
+ y14.y34
+ y14.y24
+ y18
Restriction to maximal elementary abelian number 2, which is V8
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
0
- y4 restricts to
0
- u1 restricts to
y12.y23
+ y14.y2
- u2 restricts to
0
- r restricts to
y12.y22.y34
+ y12.y23.y33
+ y12.y25.y3
+ y14.y34
+ y14.y2.y33
+ y14.y23.y3
+ y14.y24
+ y18
Restriction to maximal elementary abelian number 3, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
0
- u1 restricts to
y22.y33
+ y24.y3
- u2 restricts to
y22.y33
- r restricts to
y38
+ y23.y35
+ y24.y34
+ y27.y3
+ y12.y22.y34
+ y12.y24.y32
+ y14.y34
+ y14.y22.y32
+ y14.y24
+ y18
Restriction to the greatest central elementary abelian, which is C2
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- y4 restricts to
0
- u1 restricts to
0
- u2 restricts to
0
- r restricts to
y8
(1 + 4t + 8t2
+ 10t3 + 8t4 + 5t5
+ 5t6 + 7t7 + 7t8
+ 4t9 + t10) /
(1 - t2) (1 - t3) (1 - t8)
Back to the groups of order 64