Cohomology of group number 462 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 4.
  • Its center has rank 2.
  • 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 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t6  −  t5  −  t4  +  2·t3  −  2·t2  +  t  −  1)
    (t  −  1)4 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-6,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 14 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. c_1_2, a Duflot regular 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. b_2_6, an element of degree 2
  8. b_3_11, an element of degree 3
  9. b_3_12, an element of degree 3
  10. a_5_25, a nilpotent element of degree 5
  11. a_5_28, a nilpotent element of degree 5
  12. a_6_37, a nilpotent element of degree 6
  13. a_6_40, a nilpotent element of degree 6
  14. c_8_75, a Duflot regular element of degree 8

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Ring relations

There are 53 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_12
  3. a_1_0·a_1_1
  4. a_2_3·a_1_0
  5. a_2_4·a_1_1
  6. a_2_4·a_1_0 + a_2_3·a_1_1
  7. b_2_6·a_1_0 + b_2_5·a_1_1
  8. a_2_32
  9. a_2_3·a_2_4
  10. a_2_42
  11. a_1_1·b_3_11 + a_2_3·b_2_6
  12. a_1_0·b_3_11 + a_2_3·b_2_5
  13. a_1_1·b_3_12 + a_2_4·b_2_6 + a_2_3·b_2_6
  14. a_1_0·b_3_12 + a_2_4·b_2_5 + a_2_3·b_2_5
  15. b_2_5·b_2_6·a_1_1 + b_2_52·a_1_1
  16. b_2_52·a_1_1 + a_2_3·b_3_11
  17. b_2_52·a_1_1 + a_2_4·b_3_11 + a_2_3·b_3_12
  18. b_2_52·a_1_1 + a_2_4·b_3_12 + a_2_4·b_3_11
  19. b_3_112 + b_2_5·b_2_62 + a_2_4·b_2_52 + a_2_3·b_2_5·b_2_6
  20. b_3_122 + b_2_5·b_2_62 + b_2_52·b_2_6 + a_2_4·b_2_5·b_2_6 + a_2_4·b_2_52
       + a_2_3·b_2_62 + a_2_3·b_2_5·b_2_6
  21. a_1_1·a_5_25
  22. a_2_3·b_2_5·b_2_6 + a_2_3·b_2_52 + a_1_0·a_5_25
  23. a_2_4·b_2_62 + a_2_4·b_2_5·b_2_6 + a_1_1·a_5_28
  24. a_2_3·b_2_5·b_2_6 + a_2_3·b_2_52 + a_1_0·a_5_28
  25. a_2_3·a_5_25
  26. b_2_6·a_5_25 + b_2_5·a_5_28 + b_2_5·a_5_25 + a_2_3·b_2_6·b_3_12
  27. a_2_3·b_2_6·b_3_12 + a_2_3·b_2_5·b_3_12 + a_2_4·a_5_25 + a_2_3·a_5_28
  28. a_2_4·a_5_28 + a_2_4·a_5_25
  29. a_6_37·a_1_1
  30. a_6_37·a_1_0 + a_2_4·a_5_25
  31. a_2_3·b_2_6·b_3_12 + a_2_3·b_2_5·b_3_12 + a_6_40·a_1_1 + a_2_4·a_5_25
  32. a_6_40·a_1_0 + a_2_4·a_5_25
  33. b_3_12·a_5_25 + b_3_11·a_5_25 + b_2_5·a_6_37 + a_2_4·b_2_53 + a_2_3·b_3_11·b_3_12
       + b_2_5·a_1_0·a_5_25
  34. b_3_12·a_5_28 + b_3_12·a_5_25 + b_3_11·a_5_28 + b_3_11·a_5_25 + b_2_6·a_6_37
       + a_2_3·b_3_11·b_3_12 + a_2_3·b_2_63 + b_2_6·a_1_1·a_5_28
  35. a_2_3·a_6_37
  36. a_2_4·a_6_37
  37. b_3_12·a_5_25 + b_2_5·a_6_40 + a_2_3·b_3_11·b_3_12 + a_2_3·b_2_53
  38. b_3_12·a_5_28 + b_3_12·a_5_25 + b_2_6·a_6_40 + a_2_3·b_3_11·b_3_12 + a_2_3·b_2_53
       + b_2_5·a_1_0·a_5_25
  39. a_2_3·a_6_40
  40. a_2_4·a_6_40
  41. a_6_37·b_3_12 + a_6_37·b_3_11 + b_2_52·a_5_28 + b_2_52·a_5_25 + a_2_3·b_2_52·b_3_11
       + a_2_4·b_2_5·a_5_25
  42. a_6_40·b_3_12 + b_2_5·b_2_6·a_5_28 + a_2_3·b_2_6·a_5_28
  43. a_6_40·b_3_11 + a_6_37·b_3_12 + b_2_5·b_2_6·a_5_28 + a_2_3·b_2_52·b_3_12
       + a_2_3·b_2_52·b_3_11 + a_2_3·b_2_6·a_5_28
  44. a_2_4·b_2_54 + a_2_3·b_2_5·b_3_11·b_3_12 + a_2_3·b_2_54 + a_5_252
       + b_2_52·a_1_0·a_5_25
  45. a_5_25·a_5_28 + a_5_252
  46. a_2_4·b_2_54 + a_2_3·b_2_64 + a_2_3·b_2_5·b_3_11·b_3_12 + a_5_282
  47. a_6_37·a_5_28 + a_6_37·a_5_25
  48. a_6_40·a_5_25 + a_6_37·a_5_25
  49. a_6_40·a_5_28 + a_2_4·b_2_52·a_5_25 + a_2_3·b_2_62·a_5_28
  50. a_6_37·a_5_25 + a_2_4·b_2_52·a_5_25 + a_2_3·c_8_75·a_1_1
  51. a_6_372
  52. a_6_402
  53. a_6_37·a_6_40

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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_1_2, a Duflot regular element of degree 1
    2. c_8_75, a Duflot regular element of degree 8
    3. b_2_62 + b_2_5·b_2_6 + b_2_52, an element of degree 4
    4. b_3_12, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 9, 12].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
  • We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 2 elements of degree 2.

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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. c_1_2 → c_1_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. b_2_6 → 0, an element of degree 2
  8. b_3_11 → 0, an element of degree 3
  9. b_3_12 → 0, an element of degree 3
  10. a_5_25 → 0, an element of degree 5
  11. a_5_28 → 0, an element of degree 5
  12. a_6_37 → 0, an element of degree 6
  13. a_6_40 → 0, an element of degree 6
  14. c_8_75 → c_1_18, an element of degree 8

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. c_1_2 → c_1_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. b_2_6 → c_1_32, an element of degree 2
  8. b_3_11 → c_1_2·c_1_32, an element of degree 3
  9. b_3_12 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  10. a_5_25 → 0, an element of degree 5
  11. a_5_28 → 0, an element of degree 5
  12. a_6_37 → 0, an element of degree 6
  13. a_6_40 → 0, an element of degree 6
  14. c_8_75 → c_1_22·c_1_36 + c_1_23·c_1_35 + c_1_12·c_1_22·c_1_34
       + c_1_12·c_1_24·c_1_32 + c_1_14·c_1_34 + c_1_14·c_1_22·c_1_32
       + c_1_14·c_1_24 + c_1_18, 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