Cohomology of group number 863 of order 128

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


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 a unique conjugacy class of maximal elementary abelian subgroups, which is 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) · (t5  +  t2  +  1)

    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 13 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_4, a nilpotent element of degree 2
  5. a_2_5, a nilpotent element of degree 2
  6. a_3_8, a nilpotent element of degree 3
  7. a_3_9, a nilpotent element of degree 3
  8. b_4_13, an element of degree 4
  9. a_5_16, a nilpotent element of degree 5
  10. a_5_18, a nilpotent element of degree 5
  11. b_6_24, an element of degree 6
  12. a_7_30, a nilpotent element of degree 7
  13. c_8_37, a Duflot regular element of degree 8

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Ring relations

There are 47 minimal relations of maximal degree 14:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_2_4·a_1_0
  4. a_2_5·a_1_1
  5. a_2_5·a_1_0 + a_2_4·a_1_1
  6. a_1_14
  7. a_2_52 + a_2_4·a_2_5 + a_2_42
  8. a_1_0·a_3_8
  9. a_1_1·a_3_9 + a_2_52
  10. a_1_0·a_3_9 + a_2_42
  11. a_2_5·a_3_8
  12. a_2_4·a_3_8
  13. a_2_4·a_3_9 + a_1_12·a_3_8
  14. b_4_13·a_1_0
  15. a_3_8·a_3_9
  16. a_3_82 + b_4_13·a_1_12 + a_1_13·a_3_8
  17. a_2_4·b_4_13 + a_3_92 + a_1_13·a_3_8
  18. a_3_82 + a_1_1·a_5_16
  19. a_1_0·a_5_16
  20. a_1_0·a_5_18 + a_1_13·a_3_8
  21. a_2_4·a_5_16 + b_4_13·a_1_13
  22. a_2_5·a_5_18 + b_4_13·a_1_13
  23. a_2_5·a_5_16 + a_2_4·a_5_18 + b_4_13·a_1_13
  24. b_6_24·a_1_1 + b_4_13·a_3_8 + a_2_5·a_5_16 + b_4_13·a_1_13
  25. b_6_24·a_1_0
  26. a_3_8·a_5_16 + b_4_13·a_1_1·a_3_8 + a_2_5·a_3_92
  27. a_2_5·a_3_92 + a_1_13·a_5_18
  28. a_2_5·b_6_24 + a_3_9·a_5_18 + a_3_9·a_5_16
  29. a_2_4·b_6_24 + a_3_9·a_5_16
  30. a_3_8·a_5_18 + a_1_1·a_7_30 + a_2_5·a_3_92
  31. a_1_0·a_7_30
  32. b_6_24·a_3_8 + b_4_132·a_1_1 + a_2_5·b_4_13·a_3_9 + b_4_13·a_1_12·a_3_8
  33. a_2_5·a_7_30 + a_2_5·b_4_13·a_3_9 + b_4_13·a_1_12·a_3_8
  34. b_6_24·a_3_9 + b_4_13·a_5_16 + b_4_132·a_1_1 + a_2_5·b_4_13·a_3_9 + a_1_12·a_7_30
  35. a_2_5·b_4_13·a_3_9 + a_2_4·a_7_30
  36. a_5_162 + b_4_13·a_3_92 + b_4_132·a_1_12
  37. a_2_5·b_4_132 + a_5_16·a_5_18 + b_4_13·a_3_92 + b_4_13·a_1_1·a_5_18
  38. a_2_5·b_4_132 + a_3_9·a_7_30
  39. a_3_8·a_7_30 + b_4_13·a_1_1·a_5_18
  40. a_2_5·b_4_132 + a_5_182 + b_4_13·a_1_1·a_5_18 + b_4_132·a_1_12 + c_8_37·a_1_12
  41. b_6_24·a_5_16 + b_4_132·a_3_9 + b_4_132·a_3_8 + b_4_13·a_1_12·a_5_18
  42. b_6_24·a_5_18 + b_4_13·a_7_30 + b_4_132·a_3_9 + a_3_92·a_5_18 + b_4_13·a_1_12·a_5_18
       + b_4_132·a_1_13 + a_2_4·c_8_37·a_1_1
  43. a_5_16·a_7_30 + b_4_13·a_3_9·a_5_18 + b_4_13·a_3_9·a_5_16 + b_4_13·a_1_1·a_7_30
  44. a_5_18·a_7_30 + b_4_13·a_3_9·a_5_16 + b_4_13·a_1_1·a_7_30 + b_4_132·a_1_1·a_3_8
       + b_4_13·a_1_13·a_5_18 + c_8_37·a_1_1·a_3_8
  45. b_6_242 + b_4_133 + b_4_13·a_3_9·a_5_16 + b_4_13·a_1_13·a_5_18 + a_2_42·c_8_37
  46. b_6_24·a_7_30 + b_4_132·a_5_18 + b_4_132·a_5_16 + b_4_133·a_1_1 + a_3_92·a_7_30
       + b_4_132·a_1_12·a_3_8
  47. a_7_302 + b_4_13·a_3_9·a_7_30 + b_4_132·a_3_92 + b_4_132·a_1_1·a_5_18
       + b_4_133·a_1_12 + b_4_13·a_1_13·a_7_30 + b_4_13·c_8_37·a_1_12
       + c_8_37·a_1_13·a_3_8


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 14.
  • 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_37, a Duflot regular element of degree 8
    3. b_4_13, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 6, 10].
  • 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_00, an element of degree 1
  2. a_1_10, an element of degree 1
  3. c_1_2c_1_0, an element of degree 1
  4. a_2_40, an element of degree 2
  5. a_2_50, an element of degree 2
  6. a_3_80, an element of degree 3
  7. a_3_90, an element of degree 3
  8. b_4_130, an element of degree 4
  9. a_5_160, an element of degree 5
  10. a_5_180, an element of degree 5
  11. b_6_240, an element of degree 6
  12. a_7_300, an element of degree 7
  13. c_8_37c_1_18, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_00, an element of degree 1
  2. a_1_10, an element of degree 1
  3. c_1_2c_1_0, an element of degree 1
  4. a_2_40, an element of degree 2
  5. a_2_50, an element of degree 2
  6. a_3_80, an element of degree 3
  7. a_3_90, an element of degree 3
  8. b_4_13c_1_24, an element of degree 4
  9. a_5_160, an element of degree 5
  10. a_5_180, an element of degree 5
  11. b_6_24c_1_26, an element of degree 6
  12. a_7_300, an element of degree 7
  13. c_8_37c_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