Cohomology of group number 87 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 16.
  • 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
    ( − 1) · (t7  +  t5  +  t3  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1)2 · (t4  +  1)
  • The a-invariants are -∞,-7,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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

Ring generators

The cohomology ring has 16 minimal generators of maximal degree 9:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. a_2_1, a nilpotent element of degree 2
  4. b_3_1, an element of degree 3
  5. b_3_2, an element of degree 3
  6. a_4_3, a nilpotent element of degree 4
  7. b_4_4, an element of degree 4
  8. a_5_4, a nilpotent element of degree 5
  9. a_5_3, a nilpotent element of degree 5
  10. b_5_6, an element of degree 5
  11. a_6_6, a nilpotent element of degree 6
  12. b_7_8, an element of degree 7
  13. b_7_9, an element of degree 7
  14. a_8_5, a nilpotent element of degree 8
  15. c_8_13, a Duflot regular element of degree 8
  16. b_9_16, 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 98 minimal relations of maximal degree 18:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_2_1·a_1_0
  4. a_2_1·b_1_1
  5. a_2_12
  6. a_1_0·b_3_1
  7. a_1_0·b_3_2
  8. a_2_1·b_3_1
  9. a_2_1·b_3_2
  10. a_4_3·a_1_0
  11. a_4_3·b_1_1
  12. b_4_4·a_1_0
  13. a_2_1·a_4_3
  14. b_3_12 + b_4_4·b_1_12 + a_2_1·b_4_4
  15. a_1_0·a_5_4
  16. b_1_1·a_5_4
  17. a_1_0·a_5_3
  18. b_1_1·a_5_3 + a_2_1·b_4_4
  19. a_1_0·b_5_6
  20. b_3_22 + b_3_12 + b_1_1·b_5_6
  21. a_4_3·b_3_1
  22. a_4_3·b_3_2
  23. a_2_1·a_5_4
  24. a_2_1·a_5_3
  25. a_2_1·b_5_6
  26. a_6_6·a_1_0
  27. a_6_6·b_1_1
  28. a_4_32
  29. b_3_1·a_5_4
  30. b_3_2·a_5_4
  31. b_3_1·a_5_3
  32. b_3_2·a_5_3 + a_4_3·b_4_4
  33. a_2_1·a_6_6
  34. a_1_0·b_7_8
  35. b_3_1·b_5_6 + b_1_1·b_7_8 + b_4_4·b_1_1·b_3_2 + a_4_3·b_4_4
  36. a_1_0·b_7_9
  37. b_3_2·b_5_6 + b_1_1·b_7_9 + b_4_4·b_1_1·b_3_2 + b_4_4·b_1_1·b_3_1
  38. a_4_3·a_5_4
  39. a_4_3·a_5_3
  40. b_4_4·a_5_4 + a_4_3·b_5_6
  41. a_6_6·b_3_1
  42. a_6_6·b_3_2 + b_4_4·a_5_4
  43. a_2_1·b_7_8
  44. b_4_4·a_5_4 + a_2_1·b_7_9
  45. a_8_5·a_1_0
  46. a_8_5·b_1_1 + b_4_4·a_5_4
  47. a_5_42
  48. a_2_1·b_4_42 + a_5_32
  49. a_5_4·a_5_3
  50. a_5_4·b_5_6
  51. a_4_3·a_6_6
  52. a_5_3·b_5_6 + b_4_4·a_6_6
  53. b_3_1·b_7_8 + b_4_4·b_3_1·b_3_2 + b_4_4·b_1_1·b_5_6 + a_5_3·b_5_6
  54. b_3_2·b_7_8 + b_3_1·b_7_9 + b_4_4·b_3_1·b_3_2 + b_4_4·b_1_1·b_5_6 + a_5_3·b_5_6
  55. b_5_62 + b_3_2·b_7_9 + b_4_4·b_3_1·b_3_2 + b_4_42·b_1_12 + a_5_3·b_5_6
       + a_2_1·b_4_42
  56. b_5_62 + b_1_13·b_7_9 + b_1_15·b_5_6 + b_4_4·b_1_1·b_5_6 + b_4_4·b_1_13·b_3_1
       + a_2_1·b_4_42 + c_8_13·b_1_12
  57. a_2_1·a_8_5
  58. a_1_0·b_9_16
  59. b_1_1·b_9_16 + b_1_13·b_7_9 + b_1_13·b_7_8 + b_1_15·b_5_6 + b_4_4·b_3_1·b_3_2
       + b_4_4·b_1_13·b_3_2 + b_4_4·b_1_16 + a_5_3·b_5_6 + a_2_1·b_4_42
  60. a_6_6·a_5_3
  61. a_6_6·a_5_4
  62. a_6_6·b_5_6
  63. a_4_3·b_7_8
  64. a_4_3·b_7_9
  65. a_8_5·b_3_1
  66. a_8_5·b_3_2
  67. a_2_1·b_9_16
  68. a_6_62
  69. a_5_3·b_7_8
  70. a_5_4·b_7_8
  71. a_5_4·b_7_9
  72. b_5_6·b_7_8 + b_1_12·b_3_1·b_7_9 + b_1_15·b_7_8 + b_4_4·b_1_1·b_7_9
       + b_4_4·b_1_1·b_7_8 + b_4_4·b_1_15·b_3_2 + b_4_42·b_1_1·b_3_1 + b_4_42·b_1_14
       + a_5_3·b_7_9 + c_8_13·b_1_1·b_3_1
  73. b_5_6·b_7_9 + b_1_17·b_5_6 + b_4_4·b_1_1·b_7_8 + b_4_4·b_1_13·b_5_6
       + b_4_4·b_1_15·b_3_2 + b_4_42·b_1_1·b_3_2 + b_4_42·b_1_14 + c_8_13·b_1_1·b_3_2
       + c_8_13·b_1_14
  74. a_4_3·a_8_5
  75. a_5_3·b_7_9 + b_4_4·a_8_5 + a_4_3·b_4_42
  76. b_3_1·b_9_16 + b_1_12·b_3_1·b_7_9 + b_1_15·b_7_8 + b_4_4·b_1_13·b_5_6
       + b_4_4·b_1_15·b_3_2 + b_4_4·b_1_15·b_3_1 + b_4_42·b_1_1·b_3_2 + a_4_3·b_4_42
  77. b_3_2·b_9_16 + b_1_12·b_3_1·b_7_9 + b_1_17·b_5_6 + b_4_4·b_1_1·b_7_8
       + b_4_4·b_1_13·b_5_6 + b_4_42·b_1_1·b_3_2 + b_4_42·b_1_1·b_3_1 + a_5_3·b_7_9
       + a_4_3·b_4_42 + c_8_13·b_1_14
  78. a_6_6·b_7_8
  79. a_6_6·b_7_9
  80. a_2_1·b_4_4·b_7_9 + a_8_5·a_5_3
  81. a_8_5·a_5_4
  82. a_8_5·b_5_6 + a_2_1·b_4_4·b_7_9
  83. a_4_3·b_9_16 + a_2_1·b_4_4·b_7_9
  84. b_7_8·b_7_9 + b_7_82 + b_1_17·b_7_8 + b_4_4·b_1_13·b_7_8 + b_4_4·b_1_14·b_3_1·b_3_2
       + b_4_4·b_1_17·b_3_2 + b_4_42·b_3_1·b_3_2 + b_4_42·b_1_13·b_3_2
       + b_4_42·b_1_13·b_3_1 + c_8_13·b_3_1·b_3_2 + c_8_13·b_1_13·b_3_1
  85. b_7_82 + b_4_4·b_1_13·b_7_9 + b_4_4·b_1_15·b_5_6 + b_4_42·b_1_13·b_3_1
       + b_4_43·b_1_12 + b_4_4·a_5_32 + b_4_4·c_8_13·b_1_12 + a_2_1·b_4_4·c_8_13
  86. b_7_92 + b_1_17·b_7_9 + b_4_4·b_1_17·b_3_2 + b_4_4·b_1_17·b_3_1
       + b_4_42·b_1_1·b_5_6 + b_4_42·b_1_16 + b_4_4·a_5_32 + c_8_13·b_1_1·b_5_6
       + c_8_13·b_1_13·b_3_2 + a_2_1·b_4_4·c_8_13
  87. a_6_6·a_8_5
  88. a_5_3·b_9_16 + b_4_42·a_6_6 + b_4_4·a_5_32
  89. a_5_4·b_9_16
  90. b_5_6·b_9_16 + b_1_14·b_3_1·b_7_9 + b_1_17·b_7_9 + b_1_17·b_7_8 + b_4_4·b_3_1·b_7_9
       + b_4_4·b_1_15·b_5_6 + b_4_4·b_1_17·b_3_1 + b_4_42·b_3_1·b_3_2 + b_4_43·b_1_12
       + c_8_13·b_1_13·b_3_2 + c_8_13·b_1_13·b_3_1 + a_2_1·b_4_4·c_8_13
  91. a_8_5·b_7_8
  92. a_8_5·b_7_9
  93. a_6_6·b_9_16
  94. a_8_52
  95. b_7_8·b_9_16 + b_1_16·b_3_1·b_7_9 + b_4_4·b_1_12·b_3_1·b_7_9 + b_4_4·b_1_15·b_7_9
       + b_4_4·b_1_15·b_7_8 + b_4_4·b_1_16·b_3_1·b_3_2 + b_4_42·b_1_1·b_7_9
       + b_4_42·b_1_1·b_7_8 + b_4_42·b_1_13·b_5_6 + b_4_42·b_1_15·b_3_2
       + b_4_42·b_1_18 + b_4_42·a_8_5 + a_4_3·b_4_43 + c_8_13·b_1_12·b_3_1·b_3_2
  96. b_7_9·b_9_16 + b_1_19·b_7_9 + b_1_19·b_7_8 + b_1_111·b_5_6
       + b_4_4·b_1_12·b_3_1·b_7_9 + b_4_4·b_1_15·b_7_9 + b_4_4·b_1_15·b_7_8
       + b_4_4·b_1_16·b_3_1·b_3_2 + b_4_4·b_1_17·b_5_6 + b_4_4·b_1_19·b_3_2
       + b_4_4·b_1_19·b_3_1 + b_4_42·b_1_1·b_7_8 + b_4_42·b_1_15·b_3_2
       + b_4_42·b_1_15·b_3_1 + b_4_43·b_1_1·b_3_1 + b_4_43·b_1_14 + b_4_42·a_8_5
       + c_8_13·b_1_12·b_3_1·b_3_2 + c_8_13·b_1_13·b_5_6 + c_8_13·b_1_15·b_3_1
       + c_8_13·b_1_18 + b_4_4·c_8_13·b_1_1·b_3_1 + a_4_3·b_4_4·c_8_13
  97. a_8_5·b_9_16
  98. b_9_162 + b_1_113·b_5_6 + b_4_4·b_1_17·b_7_9 + b_4_4·b_1_111·b_3_2
       + b_4_42·b_1_17·b_3_1 + b_4_43·b_1_1·b_5_6 + b_4_44·b_1_12 + b_4_42·a_5_32
       + c_8_13·b_1_15·b_5_6 + c_8_13·b_1_17·b_3_2 + c_8_13·b_1_110
       + b_4_4·c_8_13·b_1_16 + c_8_13·a_5_32

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_13, a Duflot regular element of degree 8
    2. b_1_14 + b_4_4, an element of degree 4
    3. b_3_1, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, 1, 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. b_1_1 → 0, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_3_1 → 0, an element of degree 3
  5. b_3_2 → 0, an element of degree 3
  6. a_4_3 → 0, an element of degree 4
  7. b_4_4 → 0, an element of degree 4
  8. a_5_4 → 0, an element of degree 5
  9. a_5_3 → 0, an element of degree 5
  10. b_5_6 → 0, an element of degree 5
  11. a_6_6 → 0, an element of degree 6
  12. b_7_8 → 0, an element of degree 7
  13. b_7_9 → 0, an element of degree 7
  14. a_8_5 → 0, an element of degree 8
  15. c_8_13 → c_1_08, an element of degree 8
  16. b_9_16 → 0, an element of degree 9

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

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_1, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_3_1 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  5. b_3_2 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
  6. a_4_3 → 0, an element of degree 4
  7. b_4_4 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  8. a_5_4 → 0, an element of degree 5
  9. a_5_3 → 0, an element of degree 5
  10. b_5_6 → c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  11. a_6_6 → 0, an element of degree 6
  12. b_7_8 → c_1_1·c_1_26 + c_1_12·c_1_25 + c_1_13·c_1_24 + c_1_14·c_1_23
       + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
       + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7
  13. b_7_9 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
       + c_1_02·c_1_14·c_1_2 + c_1_03·c_1_14 + c_1_04·c_1_1·c_1_22
       + c_1_04·c_1_12·c_1_2 + c_1_04·c_1_13 + c_1_05·c_1_12 + c_1_06·c_1_1, an element of degree 7
  14. a_8_5 → 0, an element of degree 8
  15. c_8_13 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23
       + 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_02·c_1_16 + c_1_03·c_1_15 + c_1_04·c_1_24
       + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14 + c_1_05·c_1_13 + c_1_06·c_1_12
       + c_1_08, an element of degree 8
  16. b_9_16 → c_1_1·c_1_28 + c_1_17·c_1_22 + c_1_0·c_1_12·c_1_26 + c_1_0·c_1_13·c_1_25
       + c_1_0·c_1_15·c_1_23 + c_1_0·c_1_16·c_1_22 + c_1_02·c_1_1·c_1_26
       + c_1_02·c_1_12·c_1_25 + c_1_02·c_1_14·c_1_23 + c_1_02·c_1_15·c_1_22
       + c_1_02·c_1_17 + c_1_03·c_1_16 + c_1_05·c_1_14 + c_1_06·c_1_13, 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