Small group number 513 of order 128
G is the group 128gp513
G has 3 minimal generators, rank 5 and exponent 4.
The centre has rank 2.
There are 2 conjugacy classes of maximal
elementary abelian subgroups. Their ranks are:
5, 5.
This cohomology ring calculation is complete.
Ring structure
| Completion information
| Koszul information
| Restriction information
| Poincaré series
The cohomology ring has 15 generators:
- y1 in degree 1, a nilpotent element
- y2 in degree 1
- y3 in degree 1
- x1 in degree 2
- x2 in degree 2
- x3 in degree 2
- x4 in degree 2
- x5 in degree 2
- x6 in degree 2, a regular element
- w1 in degree 3
- w2 in degree 3
- w3 in degree 3
- v1 in degree 4
- v2 in degree 4
- v3 in degree 4, a regular element
There are 53 minimal relations:
- y1.y3 =
0
- y1.y2 =
0
- y12 =
0
- y3.x5 =
0
- y3.x2 =
y1.x2
- y3.x1 =
y1.x2
- y1.x5 =
0
- y1.x4 =
0
- y1.x3 =
y1.x2
- x52 =
y22.x1
- x4.x5 =
y22.x2
- x42 =
y2.y3.x4
+ y22.x3
+ y32.x6
- x3.x5 =
x1.x4
+ y2.w3
- x3.x4 =
x1.x4
+ y3.w2
+ y2.w3
+ y2.w1
- x2.x5 =
x1.x4
- x2.x4 =
x1.x4
+ y2.w3
- x22 =
x1.x3
- y3.w3 =
0
- y1.w3 =
0
- y1.w2 =
0
- y1.w1 =
0
- x5.w3 =
y2.x1.x3
+ y2.x1.x2
- x5.w2 =
y2.v2
+ y2.x1.x4
+ y22.w3
- x5.w1 =
y2.x2.x3
+ y2.x1.x3
- x4.w3 =
y2.x2.x3
+ y2.x1.x3
- x4.w2 =
y2.v1
+ y2.y3.w2
+ y22.w1
+ y3.x3.x6
+ y1.x2.x6
- x4.w1 =
y3.v1
+ y2.x32
+ y2.x2.x3
- x3.w3 =
x2.w1
+ y1.x1.x2
- x2.w3 =
x1.w1
+ y1.x1.x2
- y3.v2 =
0
- y1.v2 =
0
- y1.v1 =
0
- w32 =
x1.x32
+ x12.x3
- w2.w3 =
x2.v1
+ x1.v1
+ y2.x2.w1
+ y2.x1.w1
- w22 =
y2.x3.w2
+ y2.x2.w2
+ y22.x1.x2
+ x32.x6
+ x1.x3.x6
+ y22.v3
- w1.w3 =
x2.x32
+ x1.x2.x3
- w1.w2 =
x3.v1
+ x2.v1
+ y2.x3.w1
+ y2.x2.w1
+ y2.y3.v3
- w12 =
x33
+ x1.x32
+ y3.x3.w1
+ y32.v3
- x5.v2 =
y2.x1.w2
+ y22.x1.x3
- x5.v1 =
y2.x2.w2
+ y22.x2.x3
+ y22.x1.x3
- x4.v2 =
y2.x2.w2
+ y22.x2.x3
- x4.v1 =
y2.x3.w2
+ y2.y3.v1
+ y22.x32
+ y22.x2.x3
+ y3.x6.w1
- x3.v2 =
x2.v1
+ x12.x4
+ y2.x1.w3
+ y2.x1.w1
- x2.v2 =
x1.v1
+ x12.x4
+ y2.x1.w3
- w3.v2 =
x1.x3.w2
+ x1.x2.w2
+ y2.x1.x32
+ y2.x1.x2.x3
- w3.v1 =
x2.x3.w2
+ x1.x3.w2
+ y2.x2.x32
+ y2.x1.x2.x3
- w2.v2 =
y2.x1.v1
+ y2.x12.x4
+ y22.x1.w1
+ x2.x6.w1
+ x1.x6.w1
+ y2.x5.v3
- w2.v1 =
y2.x12.x4
+ y22.x1.w3
+ x3.x6.w1
+ x2.x6.w1
+ y2.x4.v3
- w1.v2 =
x2.x3.w2
+ x1.x3.w2
+ y2.x2.x32
+ y2.x1.x32
- w1.v1 =
x32.w2
+ x2.x3.w2
+ y3.x3.v1
+ y2.x33
+ y2.x1.x32
+ y3.x4.v3
- v22 =
y2.x1.x3.w2
+ y2.x1.x2.w2
+ y22.x1.x32
+ y22.x12.x2
+ x1.x32.x6
+ x12.x3.x6
+ y22.x1.v3
- v1.v2 =
y2.x2.x3.w2
+ y22.x2.x32
+ y22.x1.x32
+ y22.x12.x3
+ x2.x32.x6
+ x1.x2.x3.x6
+ y22.x2.v3
- v12 =
y2.x32.w2
+ y2.x2.x3.w2
+ y2.y3.x3.v1
+ y22.x33
+ y22.x1.x32
+ y22.x1.x2.x3
+ x33.x6
+ x1.x32.x6
+ y3.x3.x6.w1
+ y2.y3.x4.v3
+ y22.x3.v3
+ y32.x6.v3
This minimal generating set constitutes a Gröbner
basis for the relations ideal.
This cohomology ring was obtained from a calculation
out to degree 13. The cohomology ring approximation
is stable from degree 8 onwards, and
Benson's tests detect stability from degree 13
onwards.
This cohomology ring has dimension 5 and depth 3.
Here is a homogeneous system of parameters:
- h1 =
x6
in degree 2
- h2 =
v3
in degree 4
- h3 =
x32
+ x1.x3
+ x12
+ y34
+ y2.w3
+ y2.y3.x3
+ y22.x3
+ y22.x2
+ y22.x1
+ y22.y32
+ y24
in degree 4
- h4 =
x1.x32
+ x12.x3
+ y32.x32
+ y2.x2.w1
+ y2.x1.w3
+ y2.x1.w1
+ y2.y3.x32
+ y2.y33.x3
+ y22.x32
+ y22.x1.x3
+ y22.x12
+ y22.y34
+ y24.x3
+ y24.x2
+ y24.x1
+ y24.y32
in degree 6
- h5 =
y3
+ y2
in degree 1
The first
3 terms h1, h2, h3 form
a regular sequence of maximum length.
The first
2 terms h1, h2 form
a complete Duflot regular sequence.
That is, their restrictions to the greatest central elementary abelian
subgroup form a regular sequence of maximal length.
Data for Benson's test:
-
Raw filter degree type:
-1, -1, -1, 5, 11, 12.
-
Filter degree type:
-1, -2, -3, -4, -5, -5.
-
α = 0
-
The system of parameters is very strongly quasi-regular.
-
The regularity conjecture is satisfied.
A basis for R/(h1, h2, h3, h4, h5) is as follows.
-
1
in degree 0
-
y2
in degree 1
-
y1
in degree 1
-
x5
in degree 2
-
x4
in degree 2
-
x3
in degree 2
-
x2
in degree 2
-
x1
in degree 2
-
y22
in degree 2
-
w3
in degree 3
-
w2
in degree 3
-
w1
in degree 3
-
y2.x4
in degree 3
-
y2.x3
in degree 3
-
y23
in degree 3
-
y1.x2
in degree 3
-
y1.x1
in degree 3
-
v2
in degree 4
-
v1
in degree 4
-
x2.x3
in degree 4
-
x1.x5
in degree 4
-
x1.x4
in degree 4
-
x1.x3
in degree 4
-
x1.x2
in degree 4
-
x12
in degree 4
-
y2.w2
in degree 4
-
y2.w1
in degree 4
-
y22.x4
in degree 4
-
y22.x3
in degree 4
-
y24
in degree 4
-
x3.w2
in degree 5
-
x3.w1
in degree 5
-
x2.w2
in degree 5
-
x2.w1
in degree 5
-
x1.w3
in degree 5
-
x1.w2
in degree 5
-
x1.w1
in degree 5
-
y2.v1
in degree 5
-
y22.w2
in degree 5
-
y22.w1
in degree 5
-
y23.x4
in degree 5
-
y23.x3
in degree 5
-
y25
in degree 5
-
y1.x1.x2
in degree 5
-
x3.v1
in degree 6
-
x2.v1
in degree 6
-
x1.v2
in degree 6
-
x1.v1
in degree 6
-
x1.x2.x3
in degree 6
-
x12.x3
in degree 6
-
x12.x2
in degree 6
-
y2.x3.w2
in degree 6
-
y2.x3.w1
in degree 6
-
y22.v1
in degree 6
-
y23.w2
in degree 6
-
y23.w1
in degree 6
-
y24.x4
in degree 6
-
y24.x3
in degree 6
-
y26
in degree 6
-
x2.x3.w2
in degree 7
-
x1.x3.w2
in degree 7
-
x1.x3.w1
in degree 7
-
x1.x2.w2
in degree 7
-
x1.x2.w1
in degree 7
-
x12.w3
in degree 7
-
x12.w2
in degree 7
-
x12.w1
in degree 7
-
y2.x3.v1
in degree 7
-
y22.x3.w1
in degree 7
-
y23.v1
in degree 7
-
y24.w2
in degree 7
-
y24.w1
in degree 7
-
y25.x4
in degree 7
-
y25.x3
in degree 7
-
x1.x3.v1
in degree 8
-
x1.x2.v1
in degree 8
-
x12.v2
in degree 8
-
x12.v1
in degree 8
-
x12.x2.x3
in degree 8
-
y22.x3.v1
in degree 8
-
y23.x3.w1
in degree 8
-
y24.v1
in degree 8
-
y25.w2
in degree 8
-
y25.w1
in degree 8
-
y26.x3
in degree 8
-
x1.x2.x3.w2
in degree 9
-
x12.x3.w2
in degree 9
-
x12.x3.w1
in degree 9
-
x12.x2.w2
in degree 9
-
x12.x2.w1
in degree 9
-
y23.x3.v1
in degree 9
-
y24.x3.w1
in degree 9
-
y25.v1
in degree 9
-
y26.w2
in degree 9
-
y26.w1
in degree 9
-
x12.x3.v1
in degree 10
-
x12.x2.v1
in degree 10
-
y24.x3.v1
in degree 10
-
y25.x3.w1
in degree 10
-
y26.v1
in degree 10
-
x12.x2.x3.w2
in degree 11
-
y25.x3.v1
in degree 11
-
y26.x3.w1
in degree 11
-
y26.x3.v1
in degree 12
A basis for AnnR/(h1, h2, h3, h4)(h5) is as follows.
-
y1
in degree 1
-
y1.x2
in degree 3
-
y1.x1
in degree 3
-
y1.x1.x2
in degree 5
-
x1.x2.w1
+ x12.w1
+ x12.x3.h
+ x12.x2.h
+ y26.h
+ x2.w1.h2
+ x1.w3.h2
+ x1.w1.h2
+ y25.h2
+ x1.x3.h3
+ x1.x2.h3
+ y24.h3
+ w3.h4
+ y23.h4
+ x5.h5
+ y22.h5
+ y2.h6
+ h7
in degree 7
-
x12.x3.w1
+ y26.w1
+ y25.w1.h
+ y26.x3.h
+ x1.x3.w1.h2
+ x12.w1.h2
+ y24.w1.h2
+ y25.x3.h2
+ y23.w1.h3
+ y24.x3.h3
+ y22.w1.h4
+ y23.x3.h4
+ x2.x3.h5
+ x1.x5.h5
+ x1.x4.h5
+ x1.x2.h5
+ y2.w1.h5
+ y22.x3.h5
+ w3.h6
+ w1.h6
+ y2.x3.h6
+ x3.h7
+ x1.h7
in degree 9
-
x12.x2.w1
+ x13.w1
+ x13.x3.h
+ x13.x2.h
+ y26.x1.h
+ x1.x2.w1.h2
+ x12.w3.h2
+ x12.w1.h2
+ y25.x1.h2
+ x12.x3.h3
+ x12.x2.h3
+ y24.x1.h3
+ x1.w3.h4
+ y23.x1.h4
+ x1.x5.h5
+ y22.x1.h5
+ y2.x1.h6
+ x1.h7
in degree 9
-
y26.x3.w1
+ y25.x3.w1.h
+ y24.x3.w1.h2
+ y26.w1.h2
+ y23.x3.w1.h3
+ y25.w1.h3
+ x1.x3.w1.h4
+ x12.w1.h4
+ y22.x3.w1.h4
+ y24.w1.h4
+ x1.x2.x3.h5
+ y2.x3.w2.h5
+ y23.w1.h5
+ y24.x4.h5
+ y26.h5
+ x3.w2.h6
+ x3.w1.h6
+ x2.w1.h6
+ x1.w3.h6
+ y22.w2.h6
+ y25.h6
+ x1.x3.h7
+ y2.w2.h7
+ y2.w1.h7
+ y22.x4.h7
+ y24.h7
+ w3.h8
+ w1.h8
+ y23.h8
+ x5.h9
+ x4.h9
+ x2.h9
+ y22.h9
+ y2.h10
+ h11
in degree 11
A basis for AnnR/(h1, h2, h3)(h4) is as follows.
-
y1
in degree 1
-
y1.x2
in degree 3
-
y1.x1
in degree 3
-
y1.x1.x2
in degree 5
Restriction to special subgroup number 1, which is 4gp2
- y1 restricts to
0
- y2 restricts to
0
- y3 restricts to
0
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
0
- x4 restricts to
0
- x5 restricts to
0
- x6 restricts to
y12
- w1 restricts to
0
- w2 restricts to
0
- w3 restricts to
0
- v1 restricts to
0
- v2 restricts to
0
- v3 restricts to
y24
Restriction to special subgroup number 2, which is 32gp51
- y1 restricts to
0
- y2 restricts to
y4
- y3 restricts to
y3
- x1 restricts to
0
- x2 restricts to
0
- x3 restricts to
y52
+ y3.y5
- x4 restricts to
y4.y5
+ y1.y3
- x5 restricts to
0
- x6 restricts to
y1.y4
+ y12
- w1 restricts to
y53
+ y32.y5
+ y2.y32
+ y22.y3
- w2 restricts to
y4.y52
+ y3.y4.y5
+ y2.y3.y4
+ y22.y4
+ y1.y52
+ y1.y3.y5
- w3 restricts to
0
- v1 restricts to
y2.y3.y4.y5
+ y22.y4.y5
+ y1.y53
+ y1.y32.y5
+ y1.y2.y32
+ y1.y22.y3
- v2 restricts to
0
- v3 restricts to
y2.y3.y52
+ y2.y32.y5
+ y22.y52
+ y22.y3.y5
+ y22.y32
+ y24
Restriction to special subgroup number 3, which is 32gp51
- y1 restricts to
0
- y2 restricts to
y3
- y3 restricts to
0
- x1 restricts to
y52
- x2 restricts to
y4.y5
- x3 restricts to
y42
- x4 restricts to
y3.y4
- x5 restricts to
y3.y5
- x6 restricts to
y1.y3
+ y12
- w1 restricts to
y42.y5
+ y43
- w2 restricts to
y3.y42
+ y2.y3.y5
+ y22.y3
+ y1.y4.y5
+ y1.y42
- w3 restricts to
y4.y52
+ y42.y5
- v1 restricts to
y3.y42.y5
+ y2.y3.y4.y5
+ y22.y3.y4
+ y1.y42.y5
+ y1.y43
- v2 restricts to
y2.y3.y52
+ y22.y3.y5
+ y1.y4.y52
+ y1.y42.y5
- v3 restricts to
y4.y53
+ y43.y5
+ y2.y4.y52
+ y2.y42.y5
+ y22.y52
+ y22.y4.y5
+ y22.y42
+ y24
(1 + 2t + 5t2
+ 8t3 + 11t4 + 14t5
+ 14t6 + 14t7 + 11t8
+ 8t9 + 5t10 + 2t11
+ t12) /
(1 - t) (1 - t2) (1 - t4)2 (1 - t6\
)
Back to the groups of order 128