Cohomology of group number 163 of order 128

About the group Ring generators Ring relations Restriction maps Back to groups 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.

About the group Ring generators Ring relations Restriction maps Back to groups of order 128

Ring generators

The cohomology ring has 3 minimal generators of maximal degree 4:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. c_4_0, a Duflot regular element of degree 4

About the group Ring generators Ring relations Restriction maps Back to groups of order 128

Ring relations

There are 2 minimal relations of maximal degree 3:

  1. a_1_0·a_1_1
  2. a_1_13 + a_1_03


About the group Ring generators Ring relations Restriction maps Back to groups of order 128

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  1. a_1_00, an element of degree 1
  2. a_1_10, an element of degree 1
  3. c_4_0c_1_04, an element of degree 4


About the group Ring generators Ring relations Restriction maps Back to groups of order 128




Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany

E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2



Last change: 25.08.2009