Cohomology of group number 156 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 2.
- 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 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
- The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 4 minimal generators of maximal degree 2:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- c_2_1, a Duflot regular element of degree 2
- c_2_2, a Duflot regular element of degree 2
Ring relations
There are 2 minimal relations of maximal degree 2:
- a_1_02
- a_1_12 + a_1_0·a_1_1
Data used for Benson′s test
- Benson′s completion test succeeded in degree 2.
- 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_1, a Duflot regular element of degree 2
- c_2_2, a Duflot regular element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 2].
- 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 2
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- c_2_1 → c_1_02, an element of degree 2
- c_2_2 → c_1_12, an element of degree 2
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