Simon King
David J. Green
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Cohomology of group number 588 of order 128
General information on the group
- The group has 3 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 2.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
1 |
| (t + 1) · (t − 1)4 · (t2 + 1) |
- The a-invariants are -∞,-∞,-4,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 10 minimal generators of maximal degree 5:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- b_1_2, an element of degree 1
- b_2_3, an element of degree 2
- b_2_4, an element of degree 2
- c_2_5, a Duflot regular element of degree 2
- b_3_9, an element of degree 3
- b_4_10, an element of degree 4
- c_4_15, a Duflot regular element of degree 4
- b_5_21, an element of degree 5
Ring relations
There are 21 minimal relations of maximal degree 10:
- a_1_12 + a_1_02
- a_1_0·a_1_1
- a_1_0·b_1_2
- b_2_3·a_1_0
- b_2_4·a_1_1
- b_2_3·b_1_22 + b_2_32 + b_2_3·a_1_1·b_1_2
- b_1_2·b_3_9 + b_2_3·b_2_4
- a_1_1·b_3_9
- b_2_3·b_3_9 + b_2_3·b_2_4·b_1_2
- b_4_10·a_1_1
- b_4_10·a_1_0
- b_3_92 + b_2_3·b_2_42 + b_2_4·a_1_0·b_3_9 + c_4_15·a_1_02
- b_1_2·b_5_21 + b_2_3·b_4_10 + b_2_33 + b_2_32·a_1_1·b_1_2
- a_1_1·b_5_21 + b_2_32·a_1_1·b_1_2
- a_1_0·b_5_21
- b_2_3·b_5_21 + b_2_3·b_4_10·b_1_2 + b_2_33·b_1_2
- b_4_10·b_3_9 + b_2_4·b_5_21 + b_2_32·b_2_4·b_1_2
- b_4_102 + b_2_42·c_2_5·b_1_22
- b_3_9·b_5_21 + b_2_3·b_2_4·b_4_10 + b_2_33·b_2_4
- b_4_10·b_5_21 + b_2_32·b_4_10·b_1_2 + b_2_3·b_2_42·c_2_5·b_1_2
- b_5_212 + b_2_35 + b_2_34·a_1_1·b_1_2 + b_2_32·b_2_42·c_2_5
Data used for Benson′s test
- Benson′s completion test succeeded in degree 10.
- The completion test was perfect: It applied in the last degree in which a generator or relation was found.
- The following is a filter regular homogeneous system of parameters:
- c_2_5, a Duflot regular element of degree 2
- c_4_15, a Duflot regular element of degree 4
- b_1_22 + b_2_4, an element of degree 2
- b_1_22, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 2, 4, 6].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 2
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- c_2_5 → c_1_02, an element of degree 2
- b_3_9 → 0, an element of degree 3
- b_4_10 → 0, an element of degree 4
- c_4_15 → c_1_14, an element of degree 4
- b_5_21 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_2_3 → 0, an element of degree 2
- b_2_4 → c_1_32 + c_1_2·c_1_3, an element of degree 2
- c_2_5 → c_1_02, an element of degree 2
- b_3_9 → 0, an element of degree 3
- b_4_10 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3, an element of degree 4
- c_4_15 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
+ c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_21 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- b_2_3 → c_1_32, an element of degree 2
- b_2_4 → c_1_2·c_1_3 + c_1_22, an element of degree 2
- c_2_5 → c_1_02, an element of degree 2
- b_3_9 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- b_4_10 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3, an element of degree 4
- c_4_15 → c_1_34 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32
+ c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_21 → c_1_35 + c_1_0·c_1_2·c_1_33 + c_1_0·c_1_22·c_1_32, an element of degree 5
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