Cohomology of group number 139 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 2 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • 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 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t7  −  t6  −  1
    (t  +  1) · (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-4,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 15 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. a_2_2, a nilpotent element of degree 2
  4. b_2_1, an element of degree 2
  5. a_3_2, a nilpotent element of degree 3
  6. b_3_3, an element of degree 3
  7. b_4_2, an element of degree 4
  8. b_4_4, an element of degree 4
  9. a_5_5, a nilpotent element of degree 5
  10. b_5_4, an element of degree 5
  11. b_6_6, an element of degree 6
  12. b_6_7, an element of degree 6
  13. a_7_5, a nilpotent element of degree 7
  14. b_8_8, an element of degree 8
  15. c_8_10, a Duflot regular element of degree 8

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

Ring relations

There are 78 minimal relations of maximal degree 16:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_2_2·a_1_1
  4. a_2_2·a_1_0
  5. b_2_1·a_1_1
  6. a_1_14
  7. a_2_2·b_2_1 + a_2_22
  8. a_1_0·a_3_2
  9. a_1_1·b_3_3 + a_1_1·a_3_2 + a_2_22
  10. a_1_0·b_3_3 + a_2_22
  11. a_2_2·a_3_2
  12. b_2_1·a_3_2
  13. a_2_2·b_3_3 + a_1_12·a_3_2
  14. b_4_2·a_1_1
  15. b_4_2·a_1_0
  16. b_4_4·a_1_0
  17. a_3_2·b_3_3 + a_3_22
  18. a_2_2·b_4_2
  19. b_3_32 + b_2_1·b_4_4 + a_3_22
  20. a_3_22 + b_4_4·a_1_12
  21. a_1_0·a_5_5
  22. a_1_1·b_5_4
  23. a_1_0·b_5_4
  24. b_4_2·a_3_2
  25. a_2_2·a_5_5 + b_4_4·a_1_13
  26. b_2_1·a_5_5 + b_4_4·a_1_13
  27. a_2_2·b_5_4 + b_4_4·a_1_13
  28. b_6_6·a_1_1 + a_1_12·a_5_5 + b_4_4·a_1_13
  29. b_4_2·b_3_3 + b_2_1·b_5_4 + b_6_6·a_1_0 + b_4_4·a_1_13
  30. b_6_7·a_1_1 + b_4_4·a_3_2 + a_1_12·a_5_5 + b_4_4·a_1_13
  31. b_6_7·a_1_0
  32. b_4_22 + b_2_12·b_4_4
  33. a_1_13·a_5_5
  34. a_3_2·b_5_4
  35. b_3_3·b_5_4 + b_4_2·b_4_4 + b_3_3·a_5_5 + a_3_2·a_5_5
  36. a_2_2·b_6_6
  37. b_3_3·a_5_5 + a_2_2·b_6_7 + a_3_2·a_5_5
  38. b_4_2·b_4_4 + b_2_1·b_6_7 + b_3_3·a_5_5 + a_3_2·a_5_5
  39. a_3_2·a_5_5 + a_1_1·a_7_5
  40. a_1_0·a_7_5
  41. b_4_2·a_5_5
  42. b_4_2·b_5_4 + b_2_1·b_4_4·b_3_3 + b_4_4·a_1_12·a_3_2
  43. b_6_7·a_3_2 + b_6_6·a_3_2 + b_4_42·a_1_1 + b_4_4·a_1_12·a_3_2
  44. b_6_7·b_3_3 + b_4_4·b_5_4 + b_4_42·a_1_1 + b_4_4·a_1_12·a_3_2
  45. a_2_2·a_7_5 + b_4_4·a_1_12·a_3_2
  46. b_2_1·a_7_5 + b_4_4·a_1_12·a_3_2
  47. b_6_6·a_3_2 + a_1_12·a_7_5 + b_4_4·a_1_12·a_3_2
  48. b_8_8·a_1_1 + b_6_6·a_3_2 + b_4_4·a_1_12·a_3_2
  49. b_8_8·a_1_0 + b_2_1·b_6_6·a_1_0
  50. a_5_5·b_5_4 + a_2_2·b_4_42
  51. b_5_42 + b_2_1·b_4_42
  52. b_4_2·b_6_7 + b_2_1·b_4_42 + a_2_2·b_4_42
  53. a_3_2·a_7_5 + b_4_4·a_1_1·a_5_5
  54. b_3_3·a_7_5 + a_2_2·b_4_42 + b_4_4·a_1_1·a_5_5
  55. a_2_2·b_4_42 + a_5_52 + b_4_4·a_1_1·a_5_5 + c_8_10·a_1_12
  56. a_2_2·b_8_8
  57. b_4_2·b_6_6 + b_2_1·b_8_8 + b_2_12·b_6_6 + b_2_13·b_4_4
  58. b_6_7·b_5_4 + b_4_42·b_3_3 + b_4_42·a_3_2 + b_4_4·a_1_12·a_5_5
  59. b_4_2·a_7_5
  60. b_6_7·a_5_5 + b_6_6·a_5_5 + b_4_4·a_7_5 + b_4_4·a_1_12·a_5_5 + b_4_42·a_1_13
  61. b_6_6·a_5_5 + c_8_10·a_1_13
  62. b_8_8·a_3_2 + b_4_4·a_1_12·a_5_5
  63. b_8_8·b_3_3 + b_6_6·b_5_4 + b_2_1·b_6_6·b_3_3 + b_2_12·b_4_4·b_3_3
       + b_4_4·a_1_12·a_5_5 + b_2_1·c_8_10·a_1_0
  64. b_5_4·a_7_5 + a_2_2·b_4_4·b_6_7
  65. a_2_2·b_4_4·b_6_7 + a_5_5·a_7_5 + b_4_4·a_1_1·a_7_5 + c_8_10·a_1_1·a_3_2
  66. b_6_72 + b_4_43 + a_2_2·b_4_4·b_6_7 + b_4_42·a_1_1·a_3_2 + a_2_22·c_8_10
  67. b_6_72 + b_6_62 + b_4_43 + b_2_12·b_8_8 + b_2_13·b_6_7 + b_2_13·b_6_6
       + b_2_14·b_4_2 + a_2_2·b_4_4·b_6_7 + b_4_42·a_1_1·a_3_2 + b_2_12·c_8_10
  68. b_6_72 + b_6_62 + b_4_43 + b_4_2·b_8_8 + b_2_1·b_4_4·b_6_6 + b_2_14·b_4_4
       + b_2_14·b_4_2 + a_2_2·b_4_4·b_6_7 + b_4_42·a_1_1·a_3_2 + b_2_12·c_8_10
  69. b_6_6·b_6_7 + b_4_4·b_8_8 + b_2_1·b_4_4·b_6_6 + b_2_12·b_4_42 + b_4_42·a_1_1·a_3_2
  70. b_6_7·a_7_5 + b_6_6·a_7_5 + b_4_42·a_5_5
  71. b_6_7·a_7_5 + b_4_42·a_5_5 + c_8_10·a_1_12·a_3_2
  72. b_8_8·a_5_5 + b_6_7·a_7_5 + b_4_42·a_5_5 + b_4_4·a_1_12·a_7_5
  73. b_8_8·b_5_4 + b_4_4·b_6_6·b_3_3 + b_2_1·b_6_6·b_5_4 + b_2_12·b_4_4·b_5_4
       + b_4_4·a_1_12·a_7_5 + b_4_42·a_1_12·a_3_2
  74. a_7_52 + b_4_4·a_5_52
  75. b_6_6·b_8_8 + b_2_13·b_8_8 + b_2_13·b_4_42 + b_2_14·b_6_6 + b_2_15·b_4_4
       + b_2_15·b_4_2 + b_2_1·b_4_2·c_8_10 + b_2_13·c_8_10
  76. b_6_7·b_8_8 + b_4_42·b_6_6 + b_2_1·b_4_4·b_8_8 + b_2_12·b_4_4·b_6_7
       + b_2_12·b_4_4·b_6_6 + b_2_13·b_4_42 + b_4_43·a_1_12
  77. b_8_8·a_7_5 + b_4_42·a_1_12·a_5_5 + b_4_4·c_8_10·a_1_13
  78. b_8_82 + b_2_12·b_4_4·b_8_8 + b_2_13·b_4_4·b_6_7 + b_2_13·b_4_4·b_6_6
       + b_2_14·b_8_8 + b_2_14·b_4_42 + b_2_15·b_6_6 + b_2_16·b_4_2 + b_2_12·b_4_4·c_8_10
       + b_2_14·c_8_10

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 16.
  • 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_10, a Duflot regular element of degree 8
    2. b_4_4 + b_2_12, an element of degree 4
    3. b_3_3, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, 4, 9, 12].
  • 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_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_2 → 0, an element of degree 2
  4. b_2_1 → 0, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. b_3_3 → 0, an element of degree 3
  7. b_4_2 → 0, an element of degree 4
  8. b_4_4 → 0, an element of degree 4
  9. a_5_5 → 0, an element of degree 5
  10. b_5_4 → 0, an element of degree 5
  11. b_6_6 → 0, an element of degree 6
  12. b_6_7 → 0, an element of degree 6
  13. a_7_5 → 0, an element of degree 7
  14. b_8_8 → 0, an element of degree 8
  15. c_8_10 → c_1_08, an element of degree 8

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_2 → 0, an element of degree 2
  4. b_2_1 → c_1_12, an element of degree 2
  5. a_3_2 → 0, an element of degree 3
  6. b_3_3 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. b_4_2 → c_1_12·c_1_22 + c_1_13·c_1_2, an element of degree 4
  8. b_4_4 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  9. a_5_5 → 0, an element of degree 5
  10. b_5_4 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  11. b_6_6 → c_1_14·c_1_22 + c_1_15·c_1_2 + c_1_0·c_1_13·c_1_22 + c_1_0·c_1_14·c_1_2
       + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_13·c_1_2 + c_1_02·c_1_14
       + c_1_04·c_1_12, an element of degree 6
  12. b_6_7 → c_1_26 + c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_13·c_1_23, an element of degree 6
  13. a_7_5 → 0, an element of degree 7
  14. b_8_8 → c_1_16·c_1_22 + c_1_17·c_1_2 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_16
       + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14, an element of degree 8
  15. c_8_10 → 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_0·c_1_13·c_1_24 + c_1_0·c_1_15·c_1_22 + c_1_02·c_1_14·c_1_22
       + c_1_02·c_1_15·c_1_2 + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14
       + c_1_08, 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