Cohomology of group number 614 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) · (t7  +  t6  +  t5  +  t4  +  2·t2  +  t  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-5,-3. 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 8:

  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. a_2_3, a nilpotent element of degree 2
  5. a_2_4, a nilpotent element of degree 2
  6. b_2_5, an element of degree 2
  7. c_2_6, a Duflot regular element of degree 2
  8. a_5_19, a nilpotent element of degree 5
  9. a_5_21, a nilpotent element of degree 5
  10. b_5_20, an element of degree 5
  11. a_6_21, a nilpotent element of degree 6
  12. c_8_45, a Duflot regular element of degree 8

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Ring relations

There are 36 minimal relations of maximal degree 12:

  1. a_1_12 + a_1_02
  2. a_1_0·a_1_1 + a_1_02
  3. a_1_0·b_1_2
  4. a_2_3·a_1_1 + a_2_3·a_1_0
  5. a_2_3·b_1_2 + a_2_4·a_1_1
  6. a_2_4·a_1_0
  7. b_2_5·b_1_2 + b_2_5·a_1_1 + b_2_5·a_1_0 + a_1_03
  8. a_2_32 + c_2_6·a_1_02
  9. a_2_3·a_2_4
  10. a_2_4·a_1_1·b_1_2 + a_2_42
  11. b_2_5·a_1_03
  12. a_1_1·a_5_19 + a_2_3·b_2_5·a_1_02
  13. a_1_0·a_5_19
  14. b_1_2·a_5_21 + b_1_2·a_5_19 + a_2_4·b_2_52 + a_2_42·b_1_22
  15. a_2_4·b_2_52 + a_1_1·a_5_21 + a_1_0·a_5_21 + a_2_3·b_2_5·a_1_02
  16. b_1_2·a_5_19 + a_1_1·b_5_20 + a_2_42·c_2_6
  17. a_1_0·b_5_20 + a_2_3·b_2_5·a_1_02
  18. a_2_3·a_5_19
  19. a_2_4·a_5_21 + a_2_4·a_5_19
  20. b_2_5·b_5_20 + b_2_53·a_1_1 + b_2_53·a_1_0 + a_2_3·b_2_52·a_1_0 + a_1_02·a_5_21
       + b_2_52·c_2_6·a_1_1 + b_2_52·c_2_6·a_1_0 + a_2_3·c_2_6·a_1_03
  21. a_2_3·b_5_20 + a_2_4·a_5_19
  22. a_1_1·b_1_2·b_5_20 + a_6_21·b_1_2 + a_2_4·b_5_20 + a_2_4·a_5_19 + a_2_42·b_1_23
       + a_2_42·c_2_6·b_1_2 + a_2_3·c_2_6·a_1_03
  23. a_6_21·a_1_1 + a_2_4·a_5_19 + a_2_3·b_2_52·a_1_0
  24. a_6_21·a_1_0 + a_2_3·b_2_52·a_1_0
  25. a_2_3·a_6_21 + a_2_3·b_2_52·a_1_02 + b_2_52·c_2_6·a_1_02
       + a_2_3·b_2_5·c_2_6·a_1_02
  26. a_2_4·a_6_21
  27. a_5_192
  28. a_5_21·b_5_20 + a_5_19·b_5_20 + b_2_52·a_1_1·a_5_21 + b_2_52·a_1_0·a_5_21
       + a_2_42·b_1_2·b_5_20 + a_2_3·b_2_5·a_1_0·a_5_21 + a_2_3·b_2_53·a_1_02
       + b_2_5·c_2_6·a_1_1·a_5_21 + b_2_5·c_2_6·a_1_0·a_5_21 + a_2_3·b_2_52·c_2_6·a_1_02
  29. b_2_52·a_6_21 + a_2_3·b_2_54 + a_5_19·a_5_21 + b_2_54·a_1_02
       + a_2_3·b_2_53·a_1_02 + b_2_53·c_2_6·a_1_02
  30. b_5_202 + b_1_25·b_5_20 + a_6_21·b_1_24 + c_8_45·b_1_22 + c_2_6·b_1_23·b_5_20
       + c_2_6·a_6_21·b_1_22 + a_2_4·c_2_6·b_1_2·b_5_20 + a_2_4·c_2_6·b_1_26
       + a_2_4·c_2_6·a_1_1·b_5_20 + a_2_42·c_2_6·b_1_24
  31. a_5_19·b_5_20 + a_6_21·b_1_24 + a_2_4·b_1_23·b_5_20 + a_2_42·b_1_26
       + a_2_3·b_2_53·a_1_02 + c_8_45·a_1_1·b_1_2 + c_2_6·a_6_21·b_1_22
       + a_2_4·c_2_6·b_1_2·b_5_20 + a_2_42·c_2_6·b_1_24 + a_2_3·b_2_52·c_2_6·a_1_02
       + a_2_42·c_2_62·b_1_22
  32. a_5_212 + b_2_52·a_1_1·a_5_21 + a_2_3·b_2_53·a_1_02 + c_8_45·a_1_02
       + b_2_53·c_2_6·a_1_02 + a_2_3·b_2_52·c_2_6·a_1_02
       + a_2_3·b_2_5·c_2_62·a_1_02 + b_2_5·c_2_63·a_1_02
  33. a_6_21·a_5_21 + a_6_21·a_5_19 + a_2_3·b_2_52·a_5_21 + b_2_52·a_1_02·a_5_21
       + b_2_5·c_2_6·a_1_02·a_5_21
  34. a_6_21·b_5_20 + a_6_21·b_1_25 + a_6_21·a_5_21 + a_2_42·b_1_22·b_5_20
       + a_2_42·b_1_27 + a_2_3·b_2_52·a_5_21 + b_2_52·a_1_02·a_5_21
       + c_8_45·a_1_1·b_1_22 + c_2_6·a_6_21·b_1_23 + a_2_4·c_8_45·b_1_2
       + a_2_42·c_2_6·b_5_20 + b_2_5·c_2_6·a_1_02·a_5_21 + a_2_3·c_2_6·a_1_02·a_5_21
       + a_2_42·c_2_62·b_1_23 + a_2_3·c_2_63·a_1_03
  35. a_6_21·a_5_21 + a_2_42·b_1_22·b_5_20 + a_2_3·b_2_52·a_5_21 + b_2_52·a_1_02·a_5_21
       + a_2_4·c_8_45·a_1_1 + a_2_42·c_2_6·b_5_20 + b_2_5·c_2_6·a_1_02·a_5_21
       + a_2_3·c_2_63·a_1_03
  36. a_6_212 + a_2_42·b_1_23·b_5_20 + b_2_54·c_2_6·a_1_02 + a_2_42·c_8_45
       + a_2_42·c_2_6·b_1_2·b_5_20

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 12.
  • 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_6, a Duflot regular element of degree 2
    2. c_8_45, a Duflot regular element of degree 8
    3. b_1_22 + b_2_5, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 9].
  • 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. a_2_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. c_2_6 → c_1_12, an element of degree 2
  8. a_5_19 → 0, an element of degree 5
  9. a_5_21 → 0, an element of degree 5
  10. b_5_20 → 0, an element of degree 5
  11. a_6_21 → 0, an element of degree 6
  12. c_8_45 → c_1_08, an element of degree 8

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. a_2_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. c_2_6 → c_1_12, an element of degree 2
  8. a_5_19 → 0, an element of degree 5
  9. a_5_21 → 0, an element of degree 5
  10. b_5_20 → c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  11. a_6_21 → 0, an element of degree 6
  12. c_8_45 → c_1_02·c_1_26 + c_1_02·c_1_12·c_1_24 + c_1_04·c_1_12·c_1_22 + c_1_08, an element of degree 8

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. a_2_3 → 0, an element of degree 2
  5. a_2_4 → 0, an element of degree 2
  6. b_2_5 → c_1_22, an element of degree 2
  7. c_2_6 → c_1_12, an element of degree 2
  8. a_5_19 → 0, an element of degree 5
  9. a_5_21 → 0, an element of degree 5
  10. b_5_20 → 0, an element of degree 5
  11. a_6_21 → 0, an element of degree 6
  12. c_8_45 → c_1_28 + c_1_12·c_1_26 + c_1_16·c_1_22 + c_1_04·c_1_24 + c_1_08, an element of degree 8

About the group Ring generators Ring relations Completion information 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