Cohomology of group number 598 of order 128

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General information on the group

  • The group has 3 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t4  +  t3  +  2·t2  +  2·t  +  1)
    (t  +  1)2 · (t  −  1)3 · (t2  +  1)
  • The a-invariants are -∞,-∞,-4,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 8 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. b_1_2, an element of degree 1
  4. b_2_3, an element of degree 2
  5. b_2_4, an element of degree 2
  6. c_2_5, a Duflot regular element of degree 2
  7. a_4_7, a nilpotent element of degree 4
  8. c_4_13, a Duflot regular element of degree 4

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Ring relations

There are 11 minimal relations of maximal degree 8:

  1. a_1_12 + a_1_02
  2. a_1_0·a_1_1
  3. a_1_0·b_1_2
  4. b_2_3·a_1_0
  5. b_2_4·b_1_2 + b_2_3·b_1_2 + b_2_4·a_1_1 + b_2_4·a_1_0 + b_2_3·a_1_1
  6. b_2_3·b_1_22 + b_2_32 + b_2_3·a_1_1·b_1_2 + c_2_5·b_1_22 + c_2_5·a_1_02
  7. a_4_7·b_1_2 + b_2_3·b_2_4·a_1_1 + b_2_32·a_1_1 + c_2_5·a_1_1·b_1_22
  8. a_4_7·a_1_1
  9. b_2_3·b_2_4·a_1_1 + b_2_32·a_1_1 + a_4_7·a_1_0
  10. b_2_3·a_4_7 + b_2_3·c_2_5·a_1_1·b_1_2
  11. a_4_72

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • 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:
    1. c_2_5, a Duflot regular element of degree 2
    2. c_4_13, a Duflot regular element of degree 4
    3. b_1_22 + b_2_4 + b_2_3, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 2, 5].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. c_2_5 → c_1_12, an element of degree 2
  7. a_4_7 → 0, an element of degree 4
  8. c_4_13 → c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_3 → c_1_1·c_1_2, an element of degree 2
  5. b_2_4 → c_1_1·c_1_2, an element of degree 2
  6. c_2_5 → c_1_1·c_1_2 + c_1_12, an element of degree 2
  7. a_4_7 → 0, an element of degree 4
  8. c_4_13 → c_1_13·c_1_2 + c_1_02·c_1_22 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → c_1_22, an element of degree 2
  6. c_2_5 → c_1_12, an element of degree 2
  7. a_4_7 → 0, an element of degree 4
  8. c_4_13 → c_1_12·c_1_22 + c_1_02·c_1_22 + c_1_04, an element of degree 4

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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