Cohomology of group number 463 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. b_5_28, an element of degree 5
  11. b_5_29, an element of degree 5
  12. b_6_37, an element of degree 6
  13. b_6_42, an 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 + b_2_5·a_1_0 + a_2_3·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 + a_2_3·b_2_5
  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_4·b_2_5
  14. a_1_0·b_3_12 + a_2_4·b_2_5
  15. b_2_5·b_2_6·a_1_1
  16. b_2_52·a_1_0 + a_2_3·b_3_11
  17. a_2_4·b_3_11 + a_2_3·b_3_12
  18. b_2_62·a_1_1 + b_2_52·a_1_1 + a_2_4·b_3_12
  19. b_3_112 + b_2_5·b_2_62 + b_2_52·b_2_6 + b_2_53 + a_2_3·b_2_5·b_2_6
  20. b_3_122 + b_2_63 + b_2_53 + a_2_4·b_2_62 + a_2_4·b_2_5·b_2_6 + a_2_4·b_2_52
       + a_2_3·b_2_52
  21. a_1_1·b_5_28 + a_2_3·b_2_62 + a_2_3·b_2_5·b_2_6
  22. a_1_0·b_5_28 + a_2_4·b_2_5·b_2_6 + a_2_3·b_2_5·b_2_6
  23. a_1_1·b_5_29 + a_2_4·b_2_62 + a_2_4·b_2_52 + a_2_3·b_2_62 + a_2_3·b_2_52
  24. a_1_0·b_5_29 + a_2_4·b_2_52 + a_2_3·b_2_52
  25. a_2_4·b_5_28 + a_2_3·b_2_6·b_3_12
  26. b_2_6·b_5_29 + b_2_6·b_5_28 + b_2_62·b_3_12 + b_2_5·b_5_29 + b_2_5·b_2_6·b_3_12
       + b_2_52·b_3_12 + b_2_52·b_3_11 + a_2_3·b_5_28 + a_2_3·b_2_6·b_3_12
       + a_2_3·b_2_5·b_3_12 + a_2_3·b_2_5·b_3_11
  27. a_2_3·b_5_29 + a_2_3·b_5_28 + a_2_3·b_2_6·b_3_12 + a_2_3·b_2_6·b_3_11
       + a_2_3·b_2_5·b_3_12 + a_2_3·b_2_5·b_3_11
  28. a_2_4·b_5_29 + a_2_4·b_2_6·b_3_12 + a_2_3·b_5_28 + a_2_3·b_2_6·b_3_12
       + a_2_3·b_2_5·b_3_12 + a_2_3·b_2_5·b_3_11
  29. b_6_37·a_1_1 + a_2_3·b_5_28 + a_2_3·b_2_5·b_3_12 + a_2_3·b_2_5·b_3_11
  30. b_6_37·a_1_0 + a_2_3·b_2_5·b_3_12 + a_2_3·b_2_5·b_3_11
  31. b_6_42·a_1_1 + a_2_4·b_2_6·b_3_12 + a_2_3·b_2_6·b_3_11 + a_2_3·b_2_5·b_3_11
  32. b_6_42·a_1_0 + a_2_3·b_5_28 + a_2_3·b_2_5·b_3_11
  33. b_3_11·b_5_29 + b_3_11·b_5_28 + b_2_6·b_3_11·b_3_12 + b_2_5·b_6_37 + b_2_53·b_2_6
       + a_2_4·b_2_52·b_2_6 + a_2_3·b_3_11·b_3_12
  34. b_3_11·b_5_29 + b_2_6·b_3_11·b_3_12 + b_2_6·b_6_37 + b_2_5·b_3_11·b_3_12 + b_2_53·b_2_6
       + b_2_54 + a_2_4·b_2_63 + a_2_4·b_2_52·b_2_6 + a_2_3·b_3_11·b_3_12 + a_2_3·b_2_53
  35. a_2_3·b_3_11·b_3_12 + a_2_3·b_6_37 + a_2_3·b_2_53
  36. a_2_4·b_6_37 + a_2_3·b_3_11·b_3_12 + a_2_3·b_2_52·b_2_6 + a_2_3·b_2_53
  37. b_3_12·b_5_29 + b_3_12·b_5_28 + b_2_64 + b_2_5·b_3_11·b_3_12 + b_2_5·b_6_42
       + b_2_5·b_2_63 + b_2_52·b_2_62 + b_2_53·b_2_6 + a_2_4·b_2_63 + a_2_3·b_2_63
       + a_2_3·b_2_53
  38. b_3_12·b_5_29 + b_2_6·b_3_11·b_3_12 + b_2_6·b_6_42 + b_2_5·b_3_11·b_3_12 + b_2_53·b_2_6
       + b_2_54 + a_2_3·b_2_63
  39. a_2_4·b_2_52·b_2_6 + a_2_3·b_6_42 + a_2_3·b_2_63 + a_2_3·b_2_53
  40. a_2_4·b_6_42 + a_2_4·b_2_63 + a_2_3·b_3_11·b_3_12
  41. b_6_37·b_3_11 + b_2_5·b_2_6·b_5_28 + b_2_52·b_5_29 + b_2_52·b_5_28
       + a_2_3·b_2_62·b_3_12 + a_2_3·b_2_52·b_3_12
  42. b_6_42·b_3_12 + b_2_62·b_5_28 + b_2_63·b_3_12 + b_2_63·b_3_11 + b_2_5·b_2_6·b_5_28
       + b_2_5·b_2_62·b_3_11 + b_2_52·b_5_28 + b_2_52·b_2_6·b_3_12 + b_2_52·b_2_6·b_3_11
       + b_2_53·b_3_12 + a_2_4·b_2_62·b_3_12 + a_2_3·b_2_5·b_2_6·b_3_12
       + a_2_3·b_2_5·b_2_6·b_3_11 + a_2_3·b_2_52·b_3_12
  43. b_6_42·b_3_11 + b_6_37·b_3_12 + b_2_63·b_3_11 + b_2_5·b_2_62·b_3_12
       + b_2_5·b_2_62·b_3_11 + b_2_53·b_3_12 + a_2_4·b_2_62·b_3_12 + a_2_3·b_2_62·b_3_12
       + a_2_3·b_2_5·b_2_6·b_3_11
  44. b_5_292 + b_5_282 + b_2_65 + b_2_54·b_2_6 + a_2_4·b_2_64 + a_2_3·b_2_5·b_6_37
       + a_2_3·b_2_53·b_2_6
  45. b_5_282 + b_2_5·b_2_64 + a_2_3·b_2_5·b_6_42 + a_2_3·b_2_53·b_2_6 + a_2_3·b_2_54
  46. b_5_28·b_5_29 + b_5_282 + b_2_62·b_3_11·b_3_12 + b_2_62·b_6_42 + b_2_65
       + b_2_5·b_2_6·b_6_42 + b_2_5·b_2_64 + b_2_52·b_3_11·b_3_12 + b_2_52·b_6_42
       + b_2_52·b_6_37 + b_2_52·b_2_63 + b_2_53·b_2_62 + b_2_54·b_2_6 + a_2_4·b_2_64
  47. b_6_37·b_5_29 + b_6_37·b_5_28 + b_2_6·b_6_37·b_3_12 + b_2_52·b_2_62·b_3_11
       + b_2_53·b_5_29 + b_2_53·b_2_6·b_3_12 + b_2_53·b_2_6·b_3_11 + b_2_54·b_3_12
       + b_2_54·b_3_11 + a_2_3·b_2_52·b_2_6·b_3_12 + a_2_3·b_2_53·b_3_12
       + a_2_3·b_2_53·b_3_11
  48. b_6_42·b_5_29 + b_6_37·b_5_29 + b_2_64·b_3_12 + b_2_64·b_3_11 + b_2_5·b_2_63·b_3_12
       + b_2_5·b_2_63·b_3_11 + b_2_54·b_3_12 + b_2_54·b_3_11 + a_2_3·b_2_63·b_3_12
       + a_2_3·b_2_52·b_2_6·b_3_12
  49. b_6_42·b_5_28 + b_2_6·b_6_37·b_3_12 + b_2_63·b_5_28 + b_2_5·b_2_63·b_3_12
       + b_2_5·b_2_63·b_3_11 + b_2_52·b_2_6·b_5_28 + b_2_52·b_2_62·b_3_12
       + b_2_52·b_2_62·b_3_11 + b_2_53·b_5_28 + b_2_53·b_2_6·b_3_12 + b_2_53·b_2_6·b_3_11
       + a_2_4·b_2_63·b_3_12 + a_2_3·b_2_63·b_3_12
  50. b_6_37·b_5_29 + b_2_6·b_6_37·b_3_12 + b_2_5·b_6_37·b_3_12 + b_2_5·b_2_63·b_3_11
       + b_2_52·b_2_62·b_3_11 + b_2_53·b_5_28 + b_2_53·b_2_6·b_3_11 + b_2_54·b_3_12
       + b_2_54·b_3_11 + a_2_3·b_2_63·b_3_12 + a_2_3·b_2_52·b_2_6·b_3_12
       + a_2_3·b_2_53·b_3_11 + a_2_3·c_8_75·a_1_1
  51. b_6_372 + b_2_52·b_2_64 + b_2_54·b_2_62 + b_2_55·b_2_6 + a_2_3·b_2_52·b_6_37
  52. b_6_422 + b_2_66 + b_2_52·b_2_64 + b_2_53·b_2_63 + b_2_56
       + a_2_3·b_2_52·b_6_42 + a_2_3·b_2_55
  53. b_6_422 + b_6_37·b_6_42 + b_2_63·b_6_37 + b_2_66 + b_2_5·b_2_62·b_6_42
       + b_2_5·b_2_62·b_6_37 + b_2_5·b_2_65 + b_2_53·b_3_11·b_3_12 + b_2_53·b_6_42
       + b_2_54·b_2_62 + b_2_56 + a_2_3·b_2_52·b_6_37 + a_2_3·b_2_54·b_2_6

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_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_2_6, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 9, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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

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. b_5_28 → 0, an element of degree 5
  11. b_5_29 → 0, an element of degree 5
  12. b_6_37 → 0, an element of degree 6
  13. b_6_42 → 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 + c_1_22, an element of degree 2
  8. b_3_11 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_23, an element of degree 3
  9. b_3_12 → c_1_33 + c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  10. b_5_28 → c_1_2·c_1_34 + c_1_25, an element of degree 5
  11. b_5_29 → c_1_35 + c_1_25, an element of degree 5
  12. b_6_37 → c_1_22·c_1_34 + c_1_24·c_1_32 + c_1_25·c_1_3 + c_1_26, an element of degree 6
  13. b_6_42 → c_1_36 + c_1_23·c_1_33 + c_1_25·c_1_3, an element of degree 6
  14. c_8_75 → c_1_2·c_1_37 + c_1_22·c_1_36 + c_1_25·c_1_33 + c_1_28
       + 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

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