Cohomology of group number 1019 of order 128

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General information on the group

  • The group has 4 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 5.
  • Its center has rank 4.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 5.


Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 5 and depth 5.
  • The depth exceeds the Duflot bound, which is 4.
  • The Poincaré series is
    ( − 1) · (t2  +  t  +  1)

    (t  +  1)2 · (t  −  1)5
  • The a-invariants are -∞,-∞,-∞,-∞,-∞,-5. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 9 minimal generators of maximal degree 2:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. c_1_3, a Duflot regular element of degree 1
  5. b_2_7, an element of degree 2
  6. b_2_8, an element of degree 2
  7. c_2_9, a Duflot regular element of degree 2
  8. c_2_10, a Duflot regular element of degree 2
  9. c_2_11, a Duflot regular element of degree 2

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Ring relations

There are 9 minimal relations of maximal degree 4:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_1_1·b_1_2 + a_1_0·b_1_2
  4. b_2_7·a_1_0
  5. b_2_7·b_1_2 + b_2_8·a_1_0
  6. b_2_8·b_1_1 + b_2_7·b_1_2
  7. b_1_14 + b_2_72 + c_2_11·b_1_12
  8. b_2_82 + b_2_8·a_1_0·b_1_2 + c_2_9·b_1_22
  9. b_2_7·b_2_8 + c_2_9·a_1_0·b_1_2


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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 5.
  • However, the last relation was already found in degree 4 and the last generator in degree 2.
  • The following is a filter regular homogeneous system of parameters:
    1. c_1_3, a Duflot regular element of degree 1
    2. c_2_9, a Duflot regular element of degree 2
    3. c_2_10, a Duflot regular element of degree 2
    4. c_2_11, a Duflot regular element of degree 2
    5. b_1_22 + b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, -1, 4].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].


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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 4

  1. a_1_00, an element of degree 1
  2. b_1_10, an element of degree 1
  3. b_1_20, an element of degree 1
  4. c_1_3c_1_0, an element of degree 1
  5. b_2_70, an element of degree 2
  6. b_2_80, an element of degree 2
  7. c_2_9c_1_32 + c_1_12, an element of degree 2
  8. c_2_10c_1_22, an element of degree 2
  9. c_2_11c_1_32, an element of degree 2

Restriction map to a maximal el. ab. subgp. of rank 5

  1. a_1_00, an element of degree 1
  2. b_1_10, an element of degree 1
  3. b_1_2c_1_4, an element of degree 1
  4. c_1_3c_1_0, an element of degree 1
  5. b_2_70, an element of degree 2
  6. b_2_8c_1_3·c_1_4 + c_1_1·c_1_4, an element of degree 2
  7. c_2_9c_1_32 + c_1_12, an element of degree 2
  8. c_2_10c_1_2·c_1_4 + c_1_22, an element of degree 2
  9. c_2_11c_1_32, an element of degree 2

Restriction map to a maximal el. ab. subgp. of rank 5

  1. a_1_00, an element of degree 1
  2. b_1_1c_1_4, an element of degree 1
  3. b_1_20, an element of degree 1
  4. c_1_3c_1_0, an element of degree 1
  5. b_2_7c_1_3·c_1_4, an element of degree 2
  6. b_2_80, an element of degree 2
  7. c_2_9c_1_3·c_1_4 + c_1_32 + c_1_1·c_1_4 + c_1_12, an element of degree 2
  8. c_2_10c_1_2·c_1_4 + c_1_22, an element of degree 2
  9. c_2_11c_1_42 + c_1_32, an element of degree 2


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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