Small group number 257 of order 64
G is the group 64gp257
The Hall-Senior number of this group is 241.
G has 4 minimal generators, rank 3 and exponent 8.
The centre has rank 1.
The 15 maximal subgroups are:
32gp38, 32gp39 (3x), 32gp42 (3x), 32gp43 (6x), E32+ (2x).
There are 6 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
3, 3, 3, 3, 3, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 5 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- y4 in degree 1
- v in degree 4, a regular element
There are 2 minimal relations:
- y1.y4 =
0
- y2.y3.y4 =
y2.y32
+ y22.y3
+ y1.y2.y3
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relation:
- y1.y2.y32 =
y1.y22.y3
+ y12.y2.y3
Essential ideal:
Zero ideal
Nilradical:
Zero ideal
This cohomology ring was obtained from a calculation
out to degree 8. The cohomology ring approximation
is stable from degree 4 onwards, and
Carlson's tests detect stability from degree 8
onwards.
This cohomology ring has dimension 3 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
v
in degree 4
- h2 =
y32
+ y22
in degree 2
- h3 =
y42
+ y2.y3
+ y12
in degree 2
The first
3 terms h1, h2, h3 form
a regular sequence of maximum length.
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 8.
-
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
-
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
-
y22.y4
in degree 3
-
y22.y3
in degree 3
-
y23
in degree 3
-
y1.y22
in degree 3
-
y12.y3
in degree 3
-
y12.y2
in degree 3
-
y13
in degree 3
-
y23.y3
in degree 4
-
y12.y22
in degree 4
-
y13.y2
in degree 4
-
y14
in degree 4
-
y14.y2
in degree 5
Restriction to maximal subgroup number 1, which is 32gp49
- y1 restricts to
y4
+ y3
+ y2
+ y1
- y2 restricts to
y3
- y3 restricts to
y2
- y4 restricts to
0
- v restricts to
y22.y32
+ y23.y3
+ v
Restriction to maximal subgroup number 2, which is 32gp49
- y1 restricts to
0
- y2 restricts to
y4
+ y2
- y3 restricts to
y1
- y4 restricts to
y4
+ y3
- v restricts to
y12.y42
+ y12.y22
+ y13.y4
+ y13.y2
+ v
Restriction to maximal subgroup number 3, which is 32gp38
- y1 restricts to
y1
- y2 restricts to
y3
- y3 restricts to
y2
- y4 restricts to
y1
- v restricts to
y23.y3
+ v
Restriction to maximal subgroup number 4, which is 32gp39
- y1 restricts to
y1
- y2 restricts to
0
- y3 restricts to
y3
- y4 restricts to
y2
- v restricts to
x2
+ y2.y3.x
+ y1.y3.x
+ y32.x
Restriction to maximal subgroup number 5, which is 32gp43
- y1 restricts to
y1
- y2 restricts to
y3
- y3 restricts to
y2
- y4 restricts to
y3
- v restricts to
y2.w
+ v
Restriction to maximal subgroup number 6, which is 32gp43
- y1 restricts to
y3
- y2 restricts to
y3
- y3 restricts to
y2
- y4 restricts to
y1
- v restricts to
y2.w
+ v
Restriction to maximal subgroup number 7, which is 32gp42
- y1 restricts to
y1
- y2 restricts to
y3
+ y1
- y3 restricts to
y2
- y4 restricts to
y3
- v restricts to
v
Restriction to maximal subgroup number 8, which is 32gp39
- y1 restricts to
y1
- y2 restricts to
y3
- y3 restricts to
0
- y4 restricts to
y2
- v restricts to
x2
+ y2.y3.x
+ y1.y3.x
+ y32.x
Restriction to maximal subgroup number 9, which is 32gp43
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y3
- y4 restricts to
y3
- v restricts to
y2.w
+ y2.y33
+ v
Restriction to maximal subgroup number 10, which is 32gp43
- y1 restricts to
y3
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y1
- v restricts to
y2.w
+ y2.y33
+ v
Restriction to maximal subgroup number 11, which is 32gp42
- y1 restricts to
y3
- y2 restricts to
y2
- y3 restricts to
y3
+ y1
- y4 restricts to
y1
- v restricts to
y2.y33
+ y13.y2
+ v
Restriction to maximal subgroup number 12, which is 32gp42
- y1 restricts to
y1
- y2 restricts to
y2
- y3 restricts to
y2
- y4 restricts to
y3
- v restricts to
v
Restriction to maximal subgroup number 13, which is 32gp43
- y1 restricts to
y3
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
- y4 restricts to
y1
- v restricts to
y2.w
+ y1.y23
+ y12.y22
+ v
Restriction to maximal subgroup number 14, which is 32gp43
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
- y3 restricts to
y2
- y4 restricts to
y3
- v restricts to
y2.w
+ y1.y23
+ y12.y22
+ v
Restriction to maximal subgroup number 15, which is 32gp39
- y1 restricts to
y1
- y2 restricts to
y2
+ y1
+ y3
- y3 restricts to
y3
- y4 restricts to
y2
- v restricts to
x2
+ y2.y3.x
+ y22.y32
+ y1.y3.x
+ y12.y32
+ y32.x
+ y2.y33
+ y1.y33
Restriction to maximal elementary abelian number 1, which is V8
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
0
- y4 restricts to
y2
- v restricts to
y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
Restriction to maximal elementary abelian number 2, which is V8
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
y3
- y4 restricts to
y2
- v restricts to
y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
Restriction to maximal elementary abelian number 3, which is V8
- y1 restricts to
y2
- y2 restricts to
y3
- y3 restricts to
0
- y4 restricts to
0
- v restricts to
y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
Restriction to maximal elementary abelian number 4, which is V8
- y1 restricts to
y2
- y2 restricts to
0
- y3 restricts to
y3
- y4 restricts to
0
- v restricts to
y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
Restriction to maximal elementary abelian number 5, which is V8
- y1 restricts to
0
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
y3
+ y2
- v restricts to
y2.y33
+ y22.y32
+ y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
Restriction to maximal elementary abelian number 6, which is V8
- y1 restricts to
y3
+ y2
- y2 restricts to
y2
- y3 restricts to
y3
- y4 restricts to
0
- v restricts to
y2.y33
+ y22.y32
+ y1.y2.y32
+ y1.y22.y3
+ y12.y32
+ y12.y2.y3
+ y12.y22
+ y14
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
- v restricts to
y4
(1 + 4t + 7t2
+ 7t3 + 4t4 + t5) /
(1 - t2)2 (1 - t4)
Back to the groups of order 64