Cohomology of group number 864 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 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

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
    ( − 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. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. b_2_4, an element of degree 2
  5. b_2_5, an element of degree 2
  6. b_3_8, an element of degree 3
  7. b_3_9, an element of degree 3
  8. b_4_13, an element of degree 4
  9. b_5_17, an element of degree 5
  10. b_5_18, an element of degree 5
  11. b_6_24, an element of degree 6
  12. b_7_30, an element of degree 7
  13. c_8_37, 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 47 minimal relations of maximal degree 14:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_1_0·b_1_22 + b_2_4·a_1_0
  4. b_2_4·b_1_1 + b_2_5·a_1_0
  5. b_2_5·b_1_1 + b_2_4·b_1_1
  6. b_1_12·b_1_22
  7. b_2_52 + b_2_4·b_1_22 + b_2_4·b_2_5 + b_2_42
  8. a_1_0·b_3_8
  9. b_2_4·b_1_22 + b_2_42 + a_1_0·b_3_9
  10. b_1_1·b_3_9 + b_1_1·b_3_8 + b_2_5·b_1_22 + b_2_52 + b_2_4·b_1_22 + b_2_42
  11. b_1_1·b_1_24 + b_2_5·b_3_8 + b_2_4·b_2_5·a_1_0
  12. b_2_4·b_3_8
  13. b_1_22·b_3_9 + b_1_1·b_1_24 + b_2_4·b_3_9 + b_2_4·b_2_5·a_1_0
  14. b_4_13·a_1_0 + b_2_4·b_2_5·a_1_0
  15. b_3_8·b_3_9 + b_3_82 + b_1_26 + b_2_5·b_1_24 + b_2_42·b_2_5 + b_2_43
       + b_2_4·a_1_0·b_3_9
  16. b_3_92 + b_3_82 + b_1_26 + b_2_4·b_4_13 + b_2_42·b_2_5 + b_2_43 + b_2_4·a_1_0·b_3_9
  17. b_3_82 + b_1_26 + b_4_13·b_1_12 + b_2_43 + b_2_4·a_1_0·b_3_9
  18. a_1_0·b_5_17
  19. a_1_0·b_5_18 + b_2_5·a_1_0·b_3_9 + b_2_4·a_1_0·b_3_9
  20. b_1_1·b_5_18 + b_1_1·b_5_17 + b_1_13·b_3_8 + b_2_5·b_1_24 + b_2_42·b_2_5
  21. b_4_13·b_1_1·b_1_22 + b_2_5·b_5_17 + b_2_5·b_1_22·b_3_8 + b_2_4·b_5_17
       + b_2_42·b_2_5·a_1_0
  22. b_4_13·b_1_1·b_1_22 + b_2_5·b_5_18
  23. b_1_22·b_5_18 + b_1_22·b_5_17 + b_1_24·b_3_8 + b_2_5·b_5_17 + b_2_5·b_1_22·b_3_8
       + b_2_4·b_5_18
  24. b_6_24·a_1_0
  25. b_1_22·b_5_17 + b_1_24·b_3_8 + b_1_12·b_5_17 + b_1_14·b_3_8 + b_6_24·b_1_1
       + b_4_13·b_3_8 + b_2_5·b_5_17
  26. b_3_8·b_5_18 + b_3_8·b_5_17 + b_1_28 + b_4_13·b_1_14 + b_2_5·b_4_13·b_1_22
       + b_2_4·b_2_5·b_4_13 + b_2_44 + b_2_4·b_2_5·a_1_0·b_3_9
  27. b_3_9·b_5_17 + b_3_8·b_5_17 + b_1_28 + b_4_13·b_1_24 + b_2_5·b_6_24
       + b_2_4·b_2_5·b_4_13 + b_2_42·b_4_13 + b_2_43·b_2_5 + b_2_44 + b_2_42·a_1_0·b_3_9
  28. b_3_9·b_5_18 + b_3_9·b_5_17 + b_4_13·b_1_14 + b_2_5·b_1_26 + b_2_4·b_6_24
       + b_2_42·b_4_13
  29. b_1_13·b_5_17 + b_1_15·b_3_8 + b_6_24·b_1_12 + b_4_13·b_1_1·b_3_8
       + b_2_5·b_4_13·b_1_22 + b_2_4·b_2_5·b_4_13 + b_2_4·b_2_5·a_1_0·b_3_9
  30. a_1_0·b_7_30
  31. b_3_8·b_5_17 + b_1_28 + b_1_1·b_7_30 + b_1_13·b_5_17 + b_1_15·b_3_8 + b_4_13·b_1_24
       + b_4_13·b_1_1·b_3_8 + b_4_13·b_1_14 + b_2_5·b_1_26 + b_2_42·b_4_13 + b_2_43·b_2_5
       + b_2_44 + b_2_4·b_2_5·a_1_0·b_3_9
  32. b_6_24·b_1_1·b_1_22 + b_2_5·b_7_30 + b_2_5·b_4_13·b_3_9 + b_2_42·b_5_17
       + b_2_42·b_2_5·b_3_9 + b_2_43·b_2_5·a_1_0
  33. b_1_22·b_7_30 + b_6_24·b_3_9 + b_6_24·b_3_8 + b_4_13·b_5_18 + b_4_13·b_5_17
       + b_4_13·b_1_22·b_3_8 + b_4_13·b_1_12·b_3_8 + b_2_5·b_1_24·b_3_8
       + b_2_5·b_4_13·b_3_9 + b_2_5·b_4_13·b_3_8 + b_2_4·b_4_13·b_3_9
  34. b_2_5·b_4_13·b_3_9 + b_2_4·b_7_30 + b_2_42·b_5_17 + b_2_42·b_2_5·b_3_9
       + b_2_43·b_2_5·a_1_0
  35. b_1_12·b_7_30 + b_6_24·b_3_9 + b_6_24·b_1_1·b_1_22 + b_6_24·b_1_13 + b_4_13·b_5_18
       + b_4_13·b_5_17 + b_4_13·b_1_22·b_3_8 + b_4_13·b_1_12·b_3_8 + b_4_132·b_1_1
       + b_2_4·b_4_13·b_3_9 + b_2_42·b_5_17 + b_2_42·b_2_5·b_3_9
  36. b_5_182 + b_5_172 + b_1_210 + b_4_13·b_1_16 + b_2_4·b_4_132
       + b_2_42·b_2_5·b_4_13 + b_2_43·b_4_13 + b_2_44·b_2_5 + b_2_45 + b_2_43·a_1_0·b_3_9
  37. b_5_17·b_5_18 + b_5_172 + b_1_210 + b_6_24·b_1_1·b_3_8 + b_4_13·b_1_26
       + b_4_13·b_1_16 + b_4_132·b_1_12 + b_2_5·b_1_28 + b_2_5·b_4_132 + b_2_43·b_4_13
       + b_2_45
  38. b_5_17·b_5_18 + b_5_172 + b_3_8·b_7_30 + b_1_210 + b_6_24·b_1_24 + b_4_13·b_1_26
       + b_4_13·b_1_1·b_5_17 + b_4_13·b_1_13·b_3_8 + b_4_13·b_1_16 + b_4_132·b_1_12
       + b_2_5·b_4_132 + b_2_42·b_6_24 + b_2_43·b_4_13 + b_2_44·b_2_5 + b_2_45
  39. b_5_17·b_5_18 + b_5_172 + b_3_9·b_7_30 + b_1_210 + b_4_13·b_1_26
       + b_4_13·b_1_1·b_5_17 + b_4_13·b_1_13·b_3_8 + b_4_13·b_1_16 + b_4_132·b_1_12
       + b_2_5·b_1_28 + b_2_5·b_4_13·b_1_24 + b_2_4·b_2_5·b_6_24 + b_2_43·b_4_13 + b_2_45
  40. b_5_17·b_5_18 + b_6_24·b_1_14 + b_4_13·b_1_26 + b_4_13·b_1_1·b_5_17
       + b_4_132·b_1_22 + b_2_5·b_1_28 + b_2_5·b_6_24·b_1_22 + b_2_5·b_4_13·b_1_24
       + b_2_4·b_4_132 + b_2_4·b_2_5·b_6_24 + b_2_42·b_2_5·b_4_13 + b_2_43·b_4_13
       + b_2_44·b_2_5 + c_8_37·b_1_12
  41. b_6_24·b_5_18 + b_6_24·b_5_17 + b_6_24·b_1_22·b_3_8 + b_6_24·b_1_12·b_3_8
       + b_4_13·b_1_12·b_5_17 + b_4_13·b_1_14·b_3_8 + b_4_13·b_6_24·b_1_1 + b_4_132·b_3_9
       + b_2_5·b_4_13·b_5_17 + b_2_5·b_4_13·b_1_22·b_3_8 + b_2_4·b_4_13·b_5_18
       + b_2_42·b_7_30 + b_2_42·b_4_13·b_3_9
  42. b_6_24·b_5_18 + b_6_24·b_1_12·b_3_8 + b_6_24·b_1_15 + b_4_13·b_7_30
       + b_4_13·b_1_14·b_3_8 + b_4_132·b_3_9 + b_4_132·b_1_13 + b_2_5·b_1_26·b_3_8
       + b_2_5·b_6_24·b_3_8 + b_2_5·b_4_13·b_5_17 + b_2_4·b_4_13·b_5_18 + b_2_42·b_4_13·b_3_9
       + c_8_37·b_1_1·b_1_22 + c_8_37·b_1_13 + b_2_5·c_8_37·a_1_0
  43. b_5_18·b_7_30 + b_5_17·b_7_30 + b_6_24·b_1_26 + b_6_24·b_1_13·b_3_8
       + b_4_13·b_6_24·b_1_12 + b_4_132·b_1_1·b_3_8 + b_2_5·b_1_210
       + b_2_5·b_6_24·b_1_24 + b_2_5·b_4_13·b_1_26 + b_2_5·b_4_13·b_6_24 + b_2_43·b_6_24
       + b_2_43·b_2_5·b_4_13 + b_2_45·b_2_5
  44. b_1_212 + b_6_24·b_1_13·b_3_8 + b_6_24·b_1_16 + b_6_242 + b_4_13·b_1_15·b_3_8
       + b_4_13·b_6_24·b_1_22 + b_4_13·b_6_24·b_1_12 + b_4_132·b_1_24
       + b_4_132·b_1_1·b_3_8 + b_4_132·b_1_14 + b_4_133 + b_2_5·b_1_210
       + b_2_5·b_4_132·b_1_22 + b_2_4·b_4_13·b_6_24 + b_2_4·b_2_5·b_4_132
       + b_2_42·b_4_132 + b_2_43·b_2_5·b_4_13 + b_2_45·b_2_5 + b_2_46
       + b_2_43·b_2_5·a_1_0·b_3_9 + c_8_37·b_1_24 + c_8_37·b_1_14 + b_2_42·c_8_37
  45. b_5_18·b_7_30 + b_6_24·b_1_16 + b_4_13·b_1_15·b_3_8 + b_4_13·b_6_24·b_1_22
       + b_2_5·b_1_210 + b_2_5·b_6_24·b_1_24 + b_2_5·b_4_13·b_1_26 + b_2_4·b_4_13·b_6_24
       + b_2_42·b_2_5·b_6_24 + b_2_43·b_2_5·b_4_13 + b_2_45·b_2_5 + c_8_37·b_1_1·b_3_8
       + c_8_37·b_1_14
  46. b_1_210·b_3_8 + b_6_24·b_7_30 + b_6_24·b_1_17 + b_4_13·b_1_16·b_3_8
       + b_4_13·b_6_24·b_3_8 + b_4_132·b_5_17 + b_4_132·b_1_12·b_3_8
       + b_2_5·b_6_24·b_1_22·b_3_8 + b_2_42·b_4_13·b_5_17 + c_8_37·b_1_22·b_3_8
       + c_8_37·b_1_12·b_3_8 + c_8_37·b_1_15 + b_2_5·c_8_37·b_3_8
  47. b_7_302 + b_6_242·b_1_22 + b_6_242·b_1_12 + b_4_13·b_6_24·b_1_1·b_3_8
       + b_4_13·b_6_24·b_1_14 + b_4_132·b_1_1·b_5_17 + b_4_133·b_1_12
       + b_2_5·b_4_13·b_6_24·b_1_22 + b_2_5·b_4_132·b_1_24 + b_2_5·b_4_133
       + b_2_4·b_4_133 + b_2_4·b_2_5·b_4_13·b_6_24 + b_2_42·b_2_5·b_4_132
       + b_2_44·b_2_5·a_1_0·b_3_9 + b_4_13·c_8_37·b_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 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_8_37, a Duflot regular element of degree 8
    2. b_1_14 + b_4_13 + b_2_4·b_2_5 + b_2_42, an element of degree 4
    3. b_1_26 + b_4_13·b_1_22 + b_4_13·b_1_12 + b_2_42·b_2_5 + b_2_43, an element of degree 6
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 15].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
  • We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 2 elements of degree 4.

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. b_1_2 → 0, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_8 → 0, an element of degree 3
  7. b_3_9 → 0, an element of degree 3
  8. b_4_13 → 0, an element of degree 4
  9. b_5_17 → 0, an element of degree 5
  10. b_5_18 → 0, an element of degree 5
  11. b_6_24 → 0, an element of degree 6
  12. b_7_30 → 0, an element of degree 7
  13. c_8_37 → 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. b_1_1 → c_1_1, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_8 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. b_3_9 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_4_13 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  9. b_5_17 → c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. b_5_18 → c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  11. b_6_24 → c_1_26 + c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_14·c_1_22
       + c_1_15·c_1_2 + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  12. b_7_30 → c_1_1·c_1_26 + c_1_12·c_1_25 + c_1_14·c_1_23 + c_1_16·c_1_2
       + c_1_02·c_1_13·c_1_22 + c_1_02·c_1_14·c_1_2 + c_1_02·c_1_15
       + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2 + c_1_04·c_1_13, an element of degree 7
  13. c_8_37 → 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_16·c_1_22 + c_1_17·c_1_2 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_15·c_1_2
       + c_1_02·c_1_16 + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2 + 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. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_8 → c_1_23, an element of degree 3
  7. b_3_9 → 0, an element of degree 3
  8. b_4_13 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
  9. b_5_17 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  10. b_5_18 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  11. b_6_24 → c_1_12·c_1_24 + c_1_16 + c_1_02·c_1_24 + c_1_04·c_1_22, an element of degree 6
  12. b_7_30 → c_1_12·c_1_25 + c_1_16·c_1_2 + c_1_02·c_1_25 + c_1_04·c_1_23, an element of degree 7
  13. c_8_37 → c_1_28 + c_1_14·c_1_24 + c_1_16·c_1_22 + c_1_02·c_1_26
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_12·c_1_22
       + c_1_04·c_1_14 + 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. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_4 → c_1_22, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_3_8 → 0, an element of degree 3
  7. b_3_9 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_4_13 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  9. b_5_17 → 0, an element of degree 5
  10. b_5_18 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
  11. b_6_24 → c_1_13·c_1_23 + c_1_14·c_1_22 + c_1_15·c_1_2 + c_1_16, an element of degree 6
  12. b_7_30 → 0, an element of degree 7
  13. c_8_37 → c_1_12·c_1_26 + c_1_14·c_1_24 + 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

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 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_4 → c_1_22, an element of degree 2
  5. b_2_5 → c_1_22, an element of degree 2
  6. b_3_8 → 0, an element of degree 3
  7. b_3_9 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_4_13 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
  9. b_5_17 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  10. b_5_18 → 0, an element of degree 5
  11. b_6_24 → c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_15·c_1_2 + c_1_16, an element of degree 6
  12. b_7_30 → c_1_12·c_1_25 + c_1_13·c_1_24 + c_1_15·c_1_22 + c_1_16·c_1_2, an element of degree 7
  13. c_8_37 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22
       + 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

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