Cohomology of group number 163 of order 128
General information on the group
- The group is also known as Q128, the Quaternion group of order 128.
- The group has 2 minimal generators and exponent 64.
- It is a (generalized) quaternion group, hence, is of p-Rank 1.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 1.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 1 and depth 1.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t2 + t + 1) |
| (t − 1) · (t2 + 1) |
- The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 3 minimal generators of maximal degree 4:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- c_4_0, a Duflot regular element of degree 4
Ring relations
There are 2 minimal relations of maximal degree 3:
- a_1_0·a_1_1
- a_1_13 + a_1_03
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
- c_4_0 → c_1_04, an element of degree 4
|