Cohomology of group number 739 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 3 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. b_1_1, an 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. b_5_20, an element of degree 5
  10. b_5_21, an element of degree 5
  11. a_6_24, 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_0·b_1_1
  2. a_1_0·b_1_2
  3. b_1_1·b_1_2
  4. a_2_3·b_1_1
  5. a_2_4·b_1_2 + a_2_3·b_1_2
  6. a_2_4·a_1_0
  7. b_2_5·b_1_2 + b_2_5·b_1_1 + a_1_03
  8. a_2_32 + c_2_6·a_1_02
  9. a_2_3·a_2_4
  10. a_2_42
  11. b_2_5·a_1_03
  12. b_1_2·a_5_19 + a_2_4·b_2_52 + a_2_3·b_1_24 + a_2_3·c_2_6·b_1_22
  13. b_1_1·a_5_19 + a_2_4·b_2_52 + a_2_4·c_2_6·b_1_12
  14. a_1_0·b_5_20
  15. b_1_1·b_5_20 + a_2_3·b_2_5·a_1_02
  16. b_1_2·b_5_21 + a_2_4·b_2_52 + a_2_3·b_2_5·a_1_02
  17. a_1_0·b_5_21 + a_1_0·a_5_19 + a_2_3·c_2_6·a_1_02 + c_2_62·a_1_02
  18. a_2_4·a_5_19 + a_2_3·c_2_6·a_1_03
  19. a_2_4·b_5_20 + a_2_3·b_5_20 + a_2_3·c_2_6·a_1_03
  20. b_2_5·b_5_21 + b_2_5·b_5_20 + b_2_5·a_5_19 + a_1_02·a_5_19 + b_2_52·c_2_6·b_1_1
       + a_2_3·b_2_5·c_2_6·a_1_0 + a_2_3·c_2_6·a_1_03 + b_2_5·c_2_62·a_1_0
  21. a_2_3·b_5_21 + a_2_3·a_5_19 + a_2_3·c_2_6·a_1_03 + a_2_3·c_2_62·a_1_0
       + c_2_62·a_1_03
  22. a_6_24·b_1_2 + a_2_3·b_5_20 + a_2_3·c_2_6·a_1_03
  23. a_6_24·a_1_0 + a_2_3·b_2_52·a_1_0 + a_2_3·b_2_5·c_2_6·a_1_0 + a_2_3·c_2_62·a_1_0
       + c_2_62·a_1_03
  24. a_6_24·b_1_1 + a_2_4·b_5_21 + a_2_4·c_2_6·b_1_13 + a_2_3·c_2_6·a_1_03
       + a_2_4·c_2_62·b_1_1
  25. a_2_3·a_6_24 + 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 + b_2_5·c_2_62·a_1_02 + a_2_3·c_2_62·a_1_02
       + c_2_63·a_1_02
  26. a_2_4·a_6_24
  27. b_5_20·b_5_21 + a_5_19·b_5_20 + a_2_3·b_1_23·b_5_20 + a_2_3·c_2_6·b_1_2·b_5_20
       + a_2_3·b_2_52·c_2_6·a_1_02
  28. a_5_19·b_5_21 + a_5_19·b_5_20 + a_2_3·b_1_23·b_5_20 + a_5_192
       + a_2_4·c_2_6·b_1_1·b_5_21 + a_2_4·b_2_53·c_2_6 + a_2_3·c_2_6·b_1_2·b_5_20
       + a_2_3·c_2_6·a_1_0·a_5_19 + c_2_62·a_1_0·a_5_19
  29. a_5_19·b_5_20 + b_2_52·a_6_24 + a_2_4·b_2_54 + a_2_3·b_1_23·b_5_20 + a_2_3·b_2_54
       + b_2_54·a_1_02 + a_2_4·b_2_53·c_2_6 + a_2_3·c_2_6·b_1_2·b_5_20
       + a_2_3·b_2_53·c_2_6 + b_2_53·c_2_6·a_1_02 + a_2_3·b_2_52·c_2_6·a_1_02
       + a_2_4·b_2_52·c_2_62 + a_2_3·b_2_52·c_2_62 + b_2_52·c_2_62·a_1_02
  30. b_5_202 + b_1_25·b_5_20 + a_2_3·b_1_23·b_5_20 + c_8_45·b_1_22
       + c_2_6·b_1_23·b_5_20 + a_2_3·c_2_6·b_1_26 + a_2_3·c_2_62·b_1_24
  31. a_2_4·b_2_54 + a_5_192 + b_2_52·a_1_0·a_5_19 + 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 + b_2_52·c_2_62·a_1_02
  32. b_5_212 + b_1_15·b_5_21 + a_5_192 + c_8_45·b_1_12 + c_2_6·b_1_13·b_5_21
       + a_2_4·c_2_6·b_1_16 + a_2_4·c_2_62·b_1_14 + c_2_64·a_1_02
  33. a_6_24·a_5_19 + a_2_3·b_2_52·a_5_19 + b_2_52·a_1_02·a_5_19
       + a_2_3·b_2_5·a_1_02·a_5_19 + a_2_3·b_2_5·c_2_6·a_5_19 + b_2_5·c_2_6·a_1_02·a_5_19
       + a_2_3·c_2_6·a_1_02·a_5_19 + a_2_3·c_2_62·a_5_19 + c_2_62·a_1_02·a_5_19
       + a_2_3·c_2_63·a_1_03
  34. a_6_24·b_5_20 + a_2_3·b_1_24·b_5_20 + a_2_3·c_8_45·b_1_2 + a_2_3·c_2_6·b_1_22·b_5_20
       + a_2_3·c_2_63·a_1_03
  35. a_6_24·b_5_21 + a_2_4·b_1_14·b_5_21 + a_2_3·b_2_52·a_5_19 + b_2_52·a_1_02·a_5_19
       + a_2_4·c_8_45·b_1_1 + a_2_3·b_2_5·c_2_6·a_5_19 + b_2_5·c_2_6·a_1_02·a_5_19
       + a_2_3·c_2_6·a_1_02·a_5_19 + a_2_4·c_2_62·b_5_21 + a_2_3·c_2_62·a_5_19
       + a_2_3·b_2_52·c_2_62·a_1_0 + c_2_62·a_1_02·a_5_19 + a_2_3·b_2_5·c_2_63·a_1_0
       + a_2_3·c_2_64·a_1_0
  36. a_6_242 + b_2_54·c_2_6·a_1_02 + b_2_52·c_2_63·a_1_02 + c_2_65·a_1_02

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

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_1_12 + 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. b_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_02, an element of degree 2
  8. a_5_19 → 0, an element of degree 5
  9. b_5_20 → 0, an element of degree 5
  10. b_5_21 → 0, an element of degree 5
  11. a_6_24 → 0, an element of degree 6
  12. c_8_45 → c_1_18, 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. b_1_1 → c_1_2, 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_0·c_1_2 + c_1_02, an element of degree 2
  8. a_5_19 → 0, an element of degree 5
  9. b_5_20 → 0, an element of degree 5
  10. b_5_21 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  11. a_6_24 → 0, an element of degree 6
  12. c_8_45 → c_1_12·c_1_26 + c_1_18 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_14·c_1_23
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22, 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. b_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_0·c_1_2 + c_1_02, an element of degree 2
  8. a_5_19 → 0, an element of degree 5
  9. b_5_20 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  10. b_5_21 → 0, an element of degree 5
  11. a_6_24 → 0, an element of degree 6
  12. c_8_45 → c_1_12·c_1_26 + c_1_18 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_14·c_1_23
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22, 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. b_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_02, an element of degree 2
  8. a_5_19 → 0, an element of degree 5
  9. b_5_20 → 0, an element of degree 5
  10. b_5_21 → 0, an element of degree 5
  11. a_6_24 → 0, an element of degree 6
  12. c_8_45 → c_1_28 + c_1_14·c_1_24 + c_1_18 + c_1_02·c_1_26 + c_1_04·c_1_24, an element of degree 8

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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