Small group number 258 of order 64
G is the group 64gp258
The Hall-Senior number of this group is 242.
G has 4 minimal generators, rank 3 and exponent 8.
The centre has rank 1.
There are 4 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
2, 3, 3, 3.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 6 generators:
- y1 in degree 1
- y2 in degree 1
- y3 in degree 1
- y4 in degree 1
- u in degree 5
- r in degree 8, a regular element
There are 5 minimal relations:
- y1.y4 =
0
- y43 =
y2.y3.y4
+ y2.y32
+ y22.y3
+ y1.y2.y3
- y2.y34 =
y22.y33
+ y23.y32
+ y24.y3
+ y13.y2.y3
- y4.u =
0
- u2 =
y12.y27.y3
+ y14.y3.u
+ y14.y2.u
+ y15.u
+ y16.y23.y3
+ y18.y2.y3
+ y12.r
A minimal Gröbner basis for the relations ideal
consists of this minimal generating set, together with the
following redundant relations:
- y1.y2.y32 =
y1.y22.y3
+ y12.y2.y3
- y2.y32.u =
y22.y3.u
+ y1.y2.y3.u
This cohomology ring was obtained from a calculation
out to degree 12. The cohomology ring approximation
is stable from degree 10 onwards, and
Benson's tests detect stability from degree 10
onwards.
This cohomology ring has dimension 3 and depth 2.
Here is a homogeneous system of parameters:
- h1 =
r
in degree 8
- h2 =
y32
+ y2.y3
+ y22
+ y1.y3
+ y1.y2
+ y12
in degree 2
- h3 =
y1
in degree 1
The first
2 terms h1, h2 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.
Data for Benson's test:
-
Raw filter degree type:
-1, -1, 7, 8.
-
Filter degree type:
-1, -2, -3, -3.
-
α = 0
-
The system of parameters is very strongly quasi-regular.
-
The regularity conjecture is satisfied.
A basis for R/(h1, h2, h3) is as follows.
-
1
in degree 0
-
y4
in degree 1
-
y3
in degree 1
-
y2
in degree 1
-
y42
in degree 2
-
y3.y4
in degree 2
-
y2.y4
in degree 2
-
y2.y3
in degree 2
-
y22
in degree 2
-
y3.y42
in degree 3
-
y2.y42
in degree 3
-
y2.y3.y4
in degree 3
-
y22.y4
in degree 3
-
y22.y3
in degree 3
-
y23
in degree 3
-
y2.y3.y42
in degree 4
-
y22.y42
in degree 4
-
y22.y3.y4
in degree 4
-
y23.y4
in degree 4
-
y23.y3
in degree 4
-
y24
in degree 4
-
u
in degree 5
-
y22.y3.y42
in degree 5
-
y23.y42
in degree 5
-
y23.y3.y4
in degree 5
-
y24.y4
in degree 5
-
y25
in degree 5
-
y3.u
in degree 6
-
y2.u
in degree 6
-
y23.y3.y42
in degree 6
-
y24.y42
in degree 6
-
y25.y4
in degree 6
-
y2.y3.u
in degree 7
-
y22.u
in degree 7
-
y25.y42
in degree 7
-
y22.y3.u
in degree 8
A basis for AnnR/(h1, h2)(h3) is as follows.
-
y4
in degree 1
-
y42
in degree 2
-
y3.y4
in degree 2
-
y2.y4
in degree 2
-
y3.y42
in degree 3
-
y2.y42
in degree 3
-
y2.y3.y4
in degree 3
-
y22.y4
in degree 3
-
y23
+ y22.h
+ y2.h2
in degree 3
-
y2.y3.y42
in degree 4
-
y22.y42
in degree 4
-
y22.y3.y4
in degree 4
-
y23.y4
in degree 4
-
y23.y3
+ y22.y3.h
+ y2.y3.h2
in degree 4
-
y24
+ y23.h
+ y22.h2
in degree 4
-
y22.y3.y42
in degree 5
-
y23.y42
in degree 5
-
y23.y3.y4
in degree 5
-
y24.y4
in degree 5
-
y25
+ y24.h
+ y23.h2
in degree 5
-
y23.y3.y42
in degree 6
-
y24.y42
in degree 6
-
y25.y4
in degree 6
-
y25.y42
in degree 7