Cohomology of group number 465 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. 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_6, an element of degree 2
  7. c_2_5, a Duflot regular 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_16, a nilpotent element of degree 5
  11. a_5_17, a nilpotent element of degree 5
  12. a_6_21, a nilpotent element of degree 6
  13. a_6_24, 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. a_1_1·b_1_22 + b_2_6·a_1_0
  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·a_1_1·b_1_2
  12. a_1_0·b_3_11 + a_2_3·b_1_22 + a_2_3·a_1_1·b_1_2
  13. a_1_1·b_3_12 + a_2_4·b_2_6
  14. a_1_0·b_3_12 + a_2_4·b_1_22
  15. b_2_6·a_1_0·b_1_22 + b_2_62·a_1_0
  16. a_2_3·b_3_11 + b_2_6·c_2_5·a_1_1
  17. a_2_4·b_3_11 + a_2_3·b_3_12
  18. b_2_6·a_1_0·b_1_22 + a_2_4·b_3_12
  19. b_3_122 + b_2_6·b_1_24 + b_2_62·a_1_1·b_1_2 + a_2_4·b_2_6·b_1_22
  20. b_3_112 + a_2_4·b_1_2·b_3_12 + a_2_4·b_1_24 + a_2_3·b_1_24 + a_2_3·b_2_6·b_1_22
       + b_2_62·c_2_5
  21. a_1_1·a_5_16 + a_2_3·c_2_5·a_1_1·b_1_2
  22. a_2_3·b_1_24 + a_2_3·b_2_6·b_1_22 + a_1_0·a_5_16
  23. a_2_4·b_2_6·b_1_22 + a_2_4·b_2_62 + a_1_1·a_5_17 + a_2_3·c_2_5·a_1_1·b_1_2
  24. a_2_3·b_1_24 + a_2_3·b_2_6·b_1_22 + a_1_0·a_5_17
  25. a_2_3·a_5_16
  26. b_1_22·a_5_17 + b_1_22·a_5_16 + b_2_6·a_5_16 + a_2_4·b_1_25 + a_2_3·b_1_22·b_3_12
       + a_2_3·b_2_6·b_1_23 + a_1_1·b_1_2·a_5_17 + a_2_4·c_2_5·b_3_12
       + a_2_3·b_2_6·c_2_5·b_1_2
  27. a_2_3·b_1_22·b_3_12 + a_2_3·b_2_6·b_3_12 + a_2_4·a_5_16 + a_2_3·a_5_17
  28. a_2_4·a_5_17 + a_2_4·a_5_16
  29. b_1_22·a_5_17 + b_1_22·a_5_16 + b_2_6·a_5_16 + a_2_4·b_1_25 + a_2_3·b_1_22·b_3_12
       + a_2_3·b_2_6·b_1_23 + a_6_21·a_1_1 + a_2_4·c_2_5·b_3_12 + a_2_3·b_2_6·c_2_5·b_1_2
  30. a_6_21·a_1_0 + a_2_4·a_5_16
  31. b_1_22·a_5_17 + b_1_22·a_5_16 + b_2_6·a_5_16 + a_2_4·b_1_25 + a_2_3·b_2_6·b_3_12
       + a_2_3·b_2_6·b_1_23 + a_6_24·a_1_1 + a_2_4·a_5_16 + a_2_4·c_2_5·b_3_12
       + a_2_3·b_2_6·c_2_5·b_1_2
  32. a_6_24·a_1_0 + a_2_4·a_5_16
  33. b_3_12·a_5_16 + a_6_21·b_1_22 + b_2_6·b_1_2·a_5_16 + a_2_4·b_2_6·b_1_2·b_3_12
       + a_2_3·b_2_62·b_1_22 + a_2_4·c_2_5·b_1_24 + a_2_3·c_2_5·b_1_2·b_3_12
       + a_2_3·b_2_6·c_2_5·b_1_22 + b_2_6·c_2_52·a_1_0·b_1_2
  34. b_3_12·a_5_17 + b_3_12·a_5_16 + b_2_6·b_1_2·a_5_17 + b_2_6·b_1_2·a_5_16 + b_2_6·a_6_21
       + a_2_4·b_2_63 + a_2_3·b_2_63 + b_2_6·a_1_1·a_5_17 + b_2_62·c_2_5·a_1_1·b_1_2
       + b_2_6·c_2_52·a_1_1·b_1_2
  35. a_2_3·b_1_2·a_5_17 + a_2_3·a_6_21 + a_2_3·c_2_52·a_1_1·b_1_2
  36. a_2_4·a_6_21
  37. b_3_12·a_5_16 + b_3_11·a_5_16 + a_6_24·b_1_22 + b_2_6·b_1_2·a_5_16 + a_2_4·b_2_63
       + a_2_3·b_2_6·b_1_2·b_3_12 + a_2_3·b_2_62·b_1_22 + b_2_6·a_1_1·a_5_17
       + a_2_4·b_1_2·a_5_16 + a_2_4·c_2_5·b_1_2·b_3_12 + a_2_3·c_2_5·b_1_2·b_3_12
       + a_2_3·b_2_6·c_2_5·b_1_22 + c_2_5·a_1_0·a_5_16 + b_2_6·c_2_52·a_1_1·b_1_2
  38. b_3_12·a_5_17 + b_3_12·a_5_16 + b_3_11·a_5_17 + b_3_11·a_5_16 + b_2_6·b_1_2·a_5_17
       + b_2_6·b_1_2·a_5_16 + b_2_6·a_6_24 + a_2_4·b_2_6·b_1_2·b_3_12 + a_2_3·b_2_62·b_1_22
       + b_2_6·a_1_1·a_5_17 + a_2_4·b_1_2·a_5_16 + a_2_4·c_2_5·b_1_2·b_3_12
       + a_2_3·b_2_62·c_2_5 + c_2_5·a_1_1·a_5_17 + a_2_3·c_2_52·a_1_1·b_1_2
  39. a_2_3·b_1_2·a_5_17 + a_2_3·a_6_24 + c_2_5·a_1_1·a_5_17 + a_2_3·c_2_52·a_1_1·b_1_2
  40. a_2_4·a_6_24
  41. a_6_21·b_3_12 + b_2_6·b_1_22·a_5_16 + b_2_6·a_6_21·b_1_2 + b_2_62·a_5_16
       + a_2_4·b_2_62·b_3_12 + a_2_4·b_2_63·b_1_2 + a_2_3·b_2_62·b_3_12
       + a_2_3·b_2_63·b_1_2 + b_2_6·a_6_21·a_1_1 + a_2_4·b_2_6·c_2_5·b_3_12
       + a_2_4·b_2_62·c_2_5·b_1_2 + a_2_3·b_2_6·c_2_5·b_1_23 + a_2_3·b_2_62·c_2_5·b_1_2
       + a_2_4·c_2_52·b_3_12 + a_2_4·b_2_6·c_2_52·b_1_2
  42. a_6_24·b_3_11 + a_6_21·b_3_11 + a_2_3·b_2_62·b_3_12 + a_2_3·b_2_62·b_1_23
       + a_2_4·b_1_22·a_5_16 + b_2_6·c_2_5·a_5_17 + b_2_6·c_2_5·a_5_16 + b_2_63·c_2_5·a_1_1
       + a_2_4·b_2_6·c_2_5·b_3_12 + a_2_3·b_2_6·c_2_5·b_1_23 + a_2_3·b_2_62·c_2_5·b_1_2
       + a_2_4·c_2_5·a_5_16 + a_2_3·c_2_5·a_5_17 + b_2_62·c_2_52·a_1_1
       + a_2_3·b_2_6·c_2_52·b_1_2
  43. a_6_24·b_3_12 + a_6_21·b_3_11 + b_2_6·b_1_22·a_5_16 + b_2_6·a_6_24·b_1_2
       + b_2_62·a_5_16 + a_2_4·b_2_62·b_3_12 + a_2_4·b_2_63·b_1_2 + a_2_3·b_2_62·b_1_23
       + b_2_63·c_2_5·a_1_1 + a_2_4·b_2_6·c_2_5·b_3_12 + a_2_4·b_2_62·c_2_5·b_1_2
       + a_2_3·b_2_6·c_2_5·b_3_12 + a_2_3·b_2_6·c_2_5·b_1_23 + a_2_3·b_2_62·c_2_5·b_1_2
       + a_2_3·b_2_6·c_2_52·b_1_2
  44. a_2_4·b_1_28 + a_2_4·b_2_64 + a_5_162 + a_1_0·b_1_24·a_5_16
       + b_2_62·a_1_1·a_5_17
  45. b_2_64·a_1_1·b_1_2 + a_2_4·b_2_62·b_1_2·b_3_12 + a_5_172 + a_5_162
  46. a_5_16·a_5_17 + a_5_162 + a_2_4·b_1_23·a_5_16 + a_2_3·c_2_5·a_6_21
       + a_2_3·c_2_53·a_1_1·b_1_2
  47. a_6_21·a_5_17 + a_6_21·a_5_16 + b_2_62·a_6_21·a_1_1 + a_2_3·b_2_62·a_5_17
       + b_2_6·c_2_5·a_6_21·a_1_1 + c_2_52·a_6_21·a_1_1
  48. a_2_3·b_2_63·b_1_23 + a_2_3·b_2_64·b_1_2 + a_6_24·a_5_17 + b_2_62·a_6_21·a_1_1
       + a_2_3·b_2_6·c_2_5·a_5_17 + c_2_52·a_6_21·a_1_1
  49. a_6_24·a_5_16 + a_6_21·a_5_17 + b_2_62·a_6_21·a_1_1 + a_2_4·b_1_24·a_5_16
       + a_2_3·b_2_62·a_5_17 + b_2_6·c_2_5·a_6_21·a_1_1 + a_2_4·c_2_5·b_1_22·a_5_16
  50. a_6_21·a_5_17 + b_2_62·a_6_21·a_1_1 + a_2_4·b_1_24·a_5_16 + a_2_3·b_2_62·a_5_17
       + b_2_6·c_2_5·a_6_21·a_1_1 + a_2_4·c_2_5·b_1_22·a_5_16 + a_2_3·c_8_75·a_1_1
       + c_2_52·a_6_21·a_1_1
  51. a_6_212
  52. a_6_242 + c_2_5·a_5_172 + c_2_5·a_5_162
  53. a_6_21·a_6_24 + b_2_62·c_2_5·a_1_1·a_5_17 + a_2_3·c_8_75·a_1_1·b_1_2
       + a_2_3·c_2_52·a_6_21 + a_2_3·c_2_54·a_1_1·b_1_2

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_5, a Duflot regular element of degree 2
    2. c_8_75, a Duflot regular element of degree 8
    3. b_1_24 + b_2_6·b_1_22 + b_2_62, an element of degree 4
    4. b_3_12 + b_2_6·b_1_2, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 10, 13].
  • 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.

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. 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_6 → 0, an element of degree 2
  7. c_2_5 → c_1_02, 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_16 → 0, an element of degree 5
  11. a_5_17 → 0, an element of degree 5
  12. a_6_21 → 0, an element of degree 6
  13. a_6_24 → 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. 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_6 → c_1_32, an element of degree 2
  7. c_2_5 → c_1_02, an element of degree 2
  8. b_3_11 → c_1_0·c_1_32, an element of degree 3
  9. b_3_12 → c_1_22·c_1_3, an element of degree 3
  10. a_5_16 → 0, an element of degree 5
  11. a_5_17 → 0, an element of degree 5
  12. a_6_21 → 0, an element of degree 6
  13. a_6_24 → 0, an element of degree 6
  14. c_8_75 → c_1_22·c_1_36 + c_1_26·c_1_32 + 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 + c_1_0·c_1_22·c_1_35 + c_1_0·c_1_23·c_1_34
       + c_1_02·c_1_36 + c_1_02·c_1_24·c_1_32 + c_1_02·c_1_25·c_1_3
       + c_1_03·c_1_2·c_1_34 + c_1_03·c_1_23·c_1_32 + c_1_04·c_1_34
       + c_1_04·c_1_22·c_1_32 + c_1_04·c_1_23·c_1_3, 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