Cohomology of group number 1587 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 4.
  • Its center has rank 3.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

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

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Ring generators

The cohomology ring has 12 minimal generators of maximal degree 5:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_1_3, a nilpotent element of degree 1
  4. b_1_2, an element of degree 1
  5. c_2_7, a Duflot regular element of degree 2
  6. a_3_10, a nilpotent element of degree 3
  7. a_3_11, a nilpotent element of degree 3
  8. b_3_12, an element of degree 3
  9. b_3_13, an element of degree 3
  10. c_4_22, a Duflot regular element of degree 4
  11. c_4_23, a Duflot regular element of degree 4
  12. b_5_37, an element of degree 5

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Ring relations

There are 28 minimal relations of maximal degree 10:

  1. a_1_0·a_1_3
  2. a_1_32 + a_1_12 + a_1_0·a_1_1 + a_1_02
  3. a_1_0·b_1_2 + a_1_12
  4. a_1_03
  5. a_1_12·b_1_2 + a_1_02·a_1_1
  6. a_1_0·a_3_10
  7. a_1_3·a_3_11
  8. a_1_0·b_3_12 + a_1_3·a_3_10 + a_1_0·a_3_11 + c_2_7·a_1_0·a_1_1
  9. b_1_2·a_3_10 + a_1_3·b_3_12 + a_1_1·a_3_10 + c_2_7·a_1_1·a_1_3
  10. a_1_0·b_3_13
  11. b_1_2·a_3_11 + a_1_3·b_3_13 + a_1_3·a_3_10 + a_1_1·a_3_11 + a_1_0·a_3_11
  12. a_1_12·b_3_12 + a_1_1·a_1_3·a_3_10 + a_1_0·a_1_1·a_3_11 + c_2_7·a_1_13
  13. a_3_10·a_3_11
  14. a_3_11·b_3_13 + c_2_7·a_1_3·b_1_23 + c_2_7·a_1_1·a_1_3·b_1_22
  15. a_3_11·b_3_12 + a_3_10·b_3_13 + a_3_112 + a_3_102 + c_2_7·a_1_1·a_3_11
  16. b_3_132 + a_1_3·b_1_22·b_3_13 + a_3_112 + a_3_102 + a_1_02·a_1_1·a_3_11
       + c_2_7·b_1_24 + c_4_22·a_1_12 + c_4_22·a_1_0·a_1_1
  17. a_3_112 + a_1_02·a_1_1·a_3_11 + c_4_22·a_1_02
  18. b_3_122 + a_1_3·b_1_22·b_3_13 + a_1_1·b_1_22·b_3_12 + a_3_112 + a_3_102
       + a_1_1·a_1_3·b_1_2·b_3_13 + c_4_23·b_1_22 + c_4_23·a_1_0·a_1_1 + c_2_72·a_1_12
  19. b_3_122 + a_1_3·b_1_22·b_3_13 + a_1_1·b_1_22·b_3_12 + a_3_112
       + a_1_1·a_1_3·b_1_2·b_3_13 + a_1_02·a_1_1·a_3_11 + c_4_23·b_1_22 + c_4_23·a_1_12
       + c_4_23·a_1_02 + c_2_72·a_1_12
  20. a_3_10·b_3_12 + a_1_1·a_1_3·b_1_2·b_3_12 + c_4_23·a_1_3·b_1_2 + c_4_23·a_1_1·a_1_3
       + c_2_7·a_1_1·a_3_10
  21. b_3_12·b_3_13 + b_1_2·b_5_37 + a_3_10·b_3_12 + a_1_1·b_5_37 + a_1_1·b_1_22·b_3_13
       + a_1_1·a_1_3·b_1_2·b_3_12 + c_2_7·b_1_2·b_3_13 + c_4_22·a_1_3·b_1_2
       + c_2_7·a_1_3·b_3_12 + c_4_22·a_1_1·a_1_3 + c_2_7·a_1_1·a_3_10 + c_2_72·a_1_3·b_1_2
  22. a_1_0·b_5_37
  23. b_3_132 + a_3_11·b_3_12 + a_1_3·b_5_37 + a_1_3·b_1_22·b_3_13 + a_3_112 + a_3_102
       + a_1_1·a_1_3·b_1_2·b_3_13 + a_1_02·a_1_1·a_3_11 + c_2_7·b_1_24 + c_2_7·a_1_3·b_3_13
       + c_2_7·a_1_3·a_3_10 + c_2_7·a_1_1·a_3_11 + c_2_72·a_1_12 + c_2_72·a_1_0·a_1_1
       + c_2_72·a_1_02
  24. a_3_11·b_5_37 + c_2_7·a_1_3·b_1_22·b_3_12 + c_2_7·a_1_1·a_1_3·b_1_24
       + c_2_72·a_1_3·b_1_23
  25. b_3_13·b_5_37 + a_3_10·b_5_37 + a_1_3·b_1_22·b_5_37 + c_2_7·b_1_23·b_3_12
       + c_4_22·a_1_3·b_3_13 + c_2_7·a_1_1·b_1_22·b_3_12 + c_2_7·a_1_1·b_1_25
       + c_2_7·a_1_02·a_1_1·a_3_11 + c_2_72·b_1_24 + c_2_72·a_1_3·b_3_13
       + c_2_72·a_1_1·b_1_23 + c_2_7·c_4_22·a_1_12 + c_2_7·c_4_22·a_1_0·a_1_1
       + c_2_7·c_4_22·a_1_02 + c_2_72·a_1_3·a_3_10
  26. b_3_12·b_5_37 + c_4_23·b_1_2·b_3_13 + c_2_7·b_1_2·b_5_37 + c_4_23·a_1_3·b_3_12
       + c_4_23·a_1_1·b_3_13 + c_4_22·a_1_3·b_3_12 + c_2_7·a_1_3·b_1_25
       + c_2_7·a_1_1·b_1_22·b_3_13 + c_2_7·a_1_1·a_1_3·b_1_2·b_3_12 + c_2_72·b_1_2·b_3_13
       + c_2_7·c_4_22·a_1_3·b_1_2 + c_2_72·a_1_1·b_3_13 + c_2_7·c_4_23·a_1_1·a_1_3
       + c_2_73·a_1_3·b_1_2 + c_2_73·a_1_1·a_1_3
  27. b_3_13·b_5_37 + a_1_3·b_1_22·b_5_37 + c_2_7·b_1_23·b_3_12 + c_4_23·a_1_3·b_3_13
       + c_4_22·a_1_3·b_3_13 + c_2_7·a_1_3·b_5_37 + c_2_7·a_1_1·b_1_22·b_3_12
       + c_2_7·a_1_1·b_1_25 + c_4_23·a_1_3·a_3_10 + c_4_22·a_1_3·a_3_10
       + c_2_7·a_1_1·a_1_3·b_1_2·b_3_13 + c_2_72·b_1_24 + c_2_72·a_1_1·b_1_23
       + c_2_72·a_1_3·a_3_10 + c_2_73·a_1_12 + c_2_73·a_1_0·a_1_1 + c_2_73·a_1_02
  28. b_5_372 + a_1_1·a_1_3·b_1_23·b_5_37 + c_4_23·a_1_3·b_1_22·b_3_13
       + c_2_7·a_1_3·b_1_24·b_3_13 + c_2_7·a_1_1·b_1_24·b_3_12
       + c_4_22·a_1_02·a_1_1·a_3_11 + c_2_7·c_4_23·b_1_24 + c_2_72·a_1_3·b_1_22·b_3_13
       + c_4_22·c_4_23·a_1_12 + c_4_22·c_4_23·a_1_0·a_1_1 + c_4_22·c_4_23·a_1_02
       + c_4_222·a_1_12 + c_4_222·a_1_0·a_1_1 + c_4_222·a_1_02 + c_2_73·b_1_24
       + c_2_72·c_4_22·a_1_12 + c_2_72·c_4_22·a_1_0·a_1_1 + c_2_72·c_4_22·a_1_02
       + c_2_74·a_1_12 + c_2_74·a_1_0·a_1_1 + c_2_74·a_1_02

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 10.
  • 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_7, a Duflot regular element of degree 2
    2. c_4_22, a Duflot regular element of degree 4
    3. c_4_23, a Duflot regular element of degree 4
    4. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 6, 8].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_3 → 0, an element of degree 1
  4. b_1_2 → 0, an element of degree 1
  5. c_2_7 → c_1_22, an element of degree 2
  6. a_3_10 → 0, an element of degree 3
  7. a_3_11 → 0, an element of degree 3
  8. b_3_12 → 0, an element of degree 3
  9. b_3_13 → 0, an element of degree 3
  10. c_4_22 → c_1_14 + c_1_04, an element of degree 4
  11. c_4_23 → c_1_04, an element of degree 4
  12. b_5_37 → 0, an element of degree 5

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_3 → 0, an element of degree 1
  4. b_1_2 → c_1_3, an element of degree 1
  5. c_2_7 → c_1_22, an element of degree 2
  6. a_3_10 → 0, an element of degree 3
  7. a_3_11 → 0, an element of degree 3
  8. b_3_12 → c_1_02·c_1_3, an element of degree 3
  9. b_3_13 → c_1_2·c_1_32, an element of degree 3
  10. c_4_22 → c_1_22·c_1_32 + c_1_12·c_1_32 + c_1_14 + c_1_02·c_1_32 + c_1_04, an element of degree 4
  11. c_4_23 → c_1_04, an element of degree 4
  12. b_5_37 → c_1_23·c_1_32 + c_1_02·c_1_2·c_1_32, an element of degree 5

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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