Cohomology of group number 935 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 1.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2 and 3, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    (2) · (t8  −  t5  −  t2  −  1/2·t  −  1/2)
    (t  +  1) · (t  −  1)3 · (t2  +  1)2 · (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 12 minimal generators of maximal degree 9:

  1. a_1_1, a nilpotent element of degree 1
  2. b_1_0, an element of degree 1
  3. b_1_2, an 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. b_4_8, an element of degree 4
  7. a_5_10, a nilpotent element of degree 5
  8. a_5_8, a nilpotent element of degree 5
  9. b_5_11, an element of degree 5
  10. b_5_12, an element of degree 5
  11. c_8_19, a Duflot regular element of degree 8
  12. b_9_23, an element of degree 9

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

Ring relations

There are 44 minimal relations of maximal degree 18:

  1. a_1_1·b_1_0
  2. b_1_0·b_1_2
  3. a_2_4·a_1_1 + a_1_13
  4. a_2_5·b_1_2
  5. a_2_4·b_1_2 + a_2_5·a_1_1 + a_1_13
  6. a_1_14
  7. a_1_12·b_1_22
  8. a_2_52 + a_2_4·a_2_5 + a_2_42
  9. b_4_8·b_1_0
  10. a_2_43
  11. b_1_0·a_5_10 + a_2_5·b_1_04 + a_2_42·b_1_02
  12. a_1_1·a_5_8 + a_1_1·a_5_10 + b_4_8·a_1_12
  13. b_1_0·a_5_8 + a_2_5·b_1_04 + a_2_4·b_1_04 + a_2_4·a_2_5·b_1_02 + a_2_42·a_2_5
  14. b_1_2·a_5_10 + a_1_1·b_5_11 + b_4_8·a_1_1·b_1_2 + a_1_1·a_5_10 + b_4_8·a_1_12
  15. b_1_0·b_5_11 + a_2_4·b_1_04 + a_2_4·a_2_5·b_1_02 + a_2_42·b_1_02
  16. b_1_2·a_5_8 + a_1_1·b_5_12 + b_4_8·a_1_1·b_1_2
  17. b_1_0·b_5_12 + a_2_4·a_2_5·b_1_02 + a_2_42·b_1_02 + a_2_42·a_2_5
  18. b_4_8·a_1_12·b_1_2 + a_2_5·b_4_8·a_1_1 + a_2_4·a_5_10 + a_2_4·a_2_5·b_1_03
       + a_1_12·a_5_10
  19. b_4_8·a_1_12·b_1_2 + a_2_5·b_4_8·a_1_1 + a_2_4·a_5_8 + a_2_4·a_2_5·b_1_03
       + a_2_42·b_1_03 + a_1_12·a_5_10 + b_4_8·a_1_13 + a_2_42·a_2_5·b_1_0
  20. a_2_5·a_5_8 + a_2_5·a_5_10 + a_2_5·b_4_8·a_1_1 + a_2_4·a_2_5·b_1_03
  21. a_2_4·b_5_11 + a_2_5·a_5_10 + a_2_5·b_4_8·a_1_1 + a_2_4·a_2_5·b_1_03
  22. a_2_5·b_5_11 + b_4_8·a_1_12·b_1_2 + a_2_5·a_5_10 + a_2_42·b_1_03
       + a_2_42·a_2_5·b_1_0
  23. a_2_4·b_5_12 + b_4_8·a_1_12·b_1_2 + a_2_5·a_5_10 + a_2_5·b_4_8·a_1_1
       + a_2_4·a_2_5·b_1_03 + a_2_42·b_1_03 + a_1_12·a_5_10
  24. a_2_5·b_5_12 + b_4_8·a_1_12·b_1_2 + a_2_5·b_4_8·a_1_1
  25. a_2_42·b_4_8 + a_1_13·a_5_10
  26. a_1_12·b_1_2·b_5_11 + a_2_4·a_2_5·b_4_8 + a_2_5·a_1_1·a_5_10
  27. a_5_82 + a_5_102 + b_4_82·a_1_12 + a_2_42·b_1_06
  28. a_5_10·a_5_8 + a_5_102 + b_4_8·a_1_1·a_5_10 + a_2_4·a_2_5·b_1_06
  29. b_5_122 + b_5_112 + b_4_8·b_1_26 + b_4_82·b_1_22 + a_1_1·b_1_24·b_5_12
       + a_1_1·b_1_24·b_5_11 + b_4_8·a_1_1·b_1_25 + a_5_102 + b_4_82·a_1_12
       + a_2_4·a_2_5·b_1_06
  30. a_5_8·b_5_12 + a_5_8·b_5_11 + a_5_10·b_5_12 + a_5_10·b_5_11 + b_4_8·a_1_1·b_1_25
       + b_4_82·a_1_1·b_1_2 + b_4_8·a_1_1·a_5_10 + b_4_82·a_1_12 + a_2_42·b_1_06
       + a_2_42·a_2_5·b_1_04
  31. a_5_8·b_5_12 + a_5_10·b_5_11 + b_4_8·a_1_1·b_5_12 + b_4_8·a_1_1·b_5_11
       + b_4_8·a_1_1·b_1_25 + b_4_82·a_1_1·b_1_2 + a_5_102 + b_4_82·a_1_12
       + a_2_42·b_1_06 + a_2_42·a_2_5·b_1_04
  32. b_5_112 + b_1_25·b_5_12 + b_1_25·b_5_11 + b_4_8·b_1_2·b_5_11 + b_4_8·b_1_26
       + a_1_1·b_1_24·b_5_11 + b_4_8·a_1_1·b_5_11 + a_2_5·b_4_82 + a_5_102
       + b_4_8·a_1_1·a_5_10 + b_4_82·a_1_12 + a_2_4·a_2_5·b_1_06 + c_8_19·b_1_22
  33. a_5_10·b_5_11 + a_1_1·b_1_24·b_5_12 + a_1_1·b_1_24·b_5_11 + b_4_8·a_1_1·b_1_25
       + a_2_5·b_4_82 + a_2_4·b_4_82 + a_5_102 + b_4_8·a_1_1·a_5_10 + a_2_42·b_1_06
       + c_8_19·a_1_1·b_1_2
  34. a_2_4·b_4_82 + a_5_102 + b_4_8·a_1_1·a_5_10 + a_2_4·a_2_5·b_1_06
       + a_2_42·b_1_06 + c_8_19·a_1_12
  35. b_5_11·b_5_12 + b_5_112 + b_1_2·b_9_23 + b_1_25·b_5_12 + b_1_25·b_5_11
       + b_4_8·b_1_2·b_5_12 + b_4_8·a_1_1·b_5_11 + b_4_82·a_1_1·b_1_2 + a_2_5·b_4_82
       + a_2_4·b_4_82 + a_5_102 + b_4_8·a_1_1·a_5_10 + a_2_4·a_2_5·b_1_06
       + a_2_42·a_2_5·b_1_04
  36. a_5_8·b_5_12 + a_5_8·b_5_11 + a_1_1·b_9_23 + a_1_1·b_1_24·b_5_12 + a_1_1·b_1_24·b_5_11
       + b_4_8·a_1_1·b_5_11 + b_4_8·a_1_1·b_1_25 + b_4_82·a_1_1·b_1_2 + a_2_4·b_4_82
       + b_4_8·a_1_1·a_5_10 + a_2_4·a_2_5·b_1_06 + a_2_42·b_1_06 + a_2_42·a_2_5·b_1_04
  37. b_1_0·b_9_23 + a_2_5·b_1_08 + a_2_4·b_1_08 + a_2_4·a_2_5·b_1_06 + a_2_42·b_1_06
  38. a_2_4·b_9_23 + a_2_5·b_4_82·a_1_1 + a_2_4·b_4_8·a_5_10 + a_2_4·a_2_5·b_1_07
       + a_2_42·b_1_07 + b_4_82·a_1_13 + a_2_42·a_2_5·b_1_05 + c_8_19·a_1_13
  39. a_2_5·b_9_23 + a_2_5·b_4_8·a_5_10 + a_2_42·b_1_07 + a_2_5·c_8_19·a_1_1
  40. a_5_8·b_9_23 + a_1_1·b_1_24·b_9_23 + a_1_1·b_1_28·b_5_12 + a_1_1·b_1_28·b_5_11
       + b_4_82·a_1_1·b_5_12 + b_4_83·a_1_1·b_1_2 + b_4_8·a_5_102 + b_4_82·a_1_1·a_5_10
       + a_2_4·a_2_5·b_1_010 + a_2_42·a_2_5·b_1_08 + c_8_19·a_1_1·b_5_12
       + c_8_19·a_1_1·b_5_11 + b_4_8·c_8_19·a_1_1·b_1_2
  41. b_5_11·b_9_23 + b_1_25·b_9_23 + b_1_29·b_5_12 + b_1_29·b_5_11 + b_4_8·b_1_25·b_5_12
       + b_4_8·b_1_210 + b_4_82·b_1_2·b_5_11 + a_5_10·b_9_23 + a_1_1·b_1_28·b_5_12
       + a_1_1·b_1_28·b_5_11 + b_4_82·a_1_1·b_5_12 + a_2_5·b_4_83 + b_4_8·a_5_102
       + a_2_4·a_2_5·b_1_010 + a_2_42·a_2_5·b_1_08 + c_8_19·b_1_2·b_5_12
       + c_8_19·b_1_2·b_5_11 + b_4_8·c_8_19·b_1_22 + c_8_19·a_1_1·b_5_12
       + c_8_19·a_1_1·b_5_11 + c_8_19·a_1_1·a_5_10 + b_4_8·c_8_19·a_1_12
  42. b_5_12·b_9_23 + b_1_25·b_9_23 + b_1_29·b_5_12 + b_1_29·b_5_11 + b_4_8·b_1_2·b_9_23
       + b_4_82·b_1_2·b_5_12 + b_4_83·b_1_22 + b_4_8·a_1_1·b_1_24·b_5_11
       + b_4_82·a_1_1·b_5_12 + b_4_82·a_1_1·b_5_11 + b_4_82·a_1_1·b_1_25
       + b_4_83·a_1_1·b_1_2 + a_2_5·b_4_83 + b_4_8·a_5_102 + b_4_83·a_1_12
       + a_2_42·a_2_5·b_1_08 + c_8_19·b_1_2·b_5_12 + c_8_19·b_1_2·b_5_11
       + b_4_8·c_8_19·b_1_22 + b_4_8·c_8_19·a_1_1·b_1_2 + b_4_8·c_8_19·a_1_12
  43. b_5_11·b_9_23 + b_1_25·b_9_23 + b_1_29·b_5_12 + b_1_29·b_5_11 + b_4_8·b_1_25·b_5_12
       + b_4_8·b_1_210 + b_4_82·b_1_2·b_5_11 + a_1_1·b_1_24·b_9_23 + b_4_8·a_1_1·b_9_23
       + b_4_8·a_1_1·b_1_24·b_5_12 + b_4_8·a_1_1·b_1_29 + b_4_82·a_1_1·b_5_12
       + b_4_82·a_1_1·b_5_11 + a_2_5·b_4_83 + b_4_83·a_1_12 + a_2_4·a_2_5·b_1_010
       + a_2_42·b_1_010 + c_8_19·b_1_2·b_5_12 + c_8_19·b_1_2·b_5_11 + b_4_8·c_8_19·b_1_22
       + b_4_8·c_8_19·a_1_1·b_1_2 + c_8_19·a_1_1·a_5_10 + b_4_8·c_8_19·a_1_12
  44. b_9_232 + b_4_8·b_1_29·b_5_12 + b_4_8·b_1_29·b_5_11 + b_4_8·b_1_214
       + b_4_82·b_1_25·b_5_11 + b_4_83·b_1_26 + b_4_84·b_1_22 + a_1_1·b_1_212·b_5_12
       + a_1_1·b_1_212·b_5_11 + b_4_8·a_1_1·b_1_24·b_9_23 + b_4_8·a_1_1·b_1_28·b_5_12
       + b_4_82·a_1_1·b_1_24·b_5_11 + b_4_83·a_1_1·b_1_25 + b_4_82·a_5_102
       + a_2_4·a_2_5·b_1_014 + b_4_8·c_8_19·b_1_26 + c_8_19·a_1_1·b_1_24·b_5_12
       + c_8_19·a_1_1·b_1_24·b_5_11 + b_4_8·c_8_19·a_1_1·b_1_25 + c_8_192·a_1_12

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 18.
  • 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_8_19, a Duflot regular element of degree 8
    2. b_1_24 + b_1_04 + b_4_8, an element of degree 4
    3. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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 1

  1. a_1_1 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_4_8 → 0, an element of degree 4
  7. a_5_10 → 0, an element of degree 5
  8. a_5_8 → 0, an element of degree 5
  9. b_5_11 → 0, an element of degree 5
  10. b_5_12 → 0, an element of degree 5
  11. c_8_19 → c_1_08, an element of degree 8
  12. b_9_23 → 0, an element of degree 9

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_1 → 0, an element of degree 1
  2. b_1_0 → c_1_1, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_4_8 → 0, an element of degree 4
  7. a_5_10 → 0, an element of degree 5
  8. a_5_8 → 0, an element of degree 5
  9. b_5_11 → 0, an element of degree 5
  10. b_5_12 → 0, an element of degree 5
  11. c_8_19 → c_1_04·c_1_14 + c_1_08, an element of degree 8
  12. b_9_23 → 0, an element of degree 9

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

  1. a_1_1 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_2 → c_1_1, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_2_5 → 0, an element of degree 2
  6. b_4_8 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  7. a_5_10 → 0, an element of degree 5
  8. a_5_8 → 0, an element of degree 5
  9. b_5_11 → c_1_1·c_1_24 + c_1_14·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. b_5_12 → c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  11. c_8_19 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_17·c_1_2
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8
  12. b_9_23 → c_1_1·c_1_28 + c_1_15·c_1_24 + c_1_17·c_1_22 + c_1_18·c_1_2
       + c_1_02·c_1_15·c_1_22 + c_1_02·c_1_16·c_1_2 + c_1_04·c_1_13·c_1_22
       + c_1_04·c_1_14·c_1_2, an element of degree 9

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