Simon King
David J. Green
Cohomology
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Cohomology of group number 130 of order 128
General information on the group
- The group has 2 minimal generators and exponent 32.
- It is non-abelian.
- It has p-Rank 2.
- Its center has rank 1.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 2 and depth 1.
- The depth coincides with the Duflot bound.
- The Poincaré series is
- The a-invariants are -∞,-2,-2. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 5 minimal generators of maximal degree 4:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- b_2_1, an element of degree 2
- a_3_1, a nilpotent element of degree 3
- c_4_2, a Duflot regular element of degree 4
Ring relations
There are 5 minimal relations of maximal degree 6:
- a_1_02
- a_1_12
- b_2_1·a_1_0
- a_1_0·a_3_1
- a_3_12
Data used for Benson′s test
- Benson′s completion test succeeded in degree 6.
- 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_4_2, a Duflot regular element of degree 4
- b_2_1, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, 2, 4].
- The filter degree type of any filter regular HSOP is [-1, -2, -2].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 1
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_2_1 → 0, an element of degree 2
- a_3_1 → 0, an element of degree 3
- c_4_2 → c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 2
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_2_1 → c_1_12, an element of degree 2
- a_3_1 → 0, an element of degree 3
- c_4_2 → c_1_02·c_1_12 + c_1_04, an element of degree 4
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