Cohomology of group number 1518 of order 256

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 256

General information on the group

  • The group has 3 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its centre has rank 2.
  • It has 5 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 4, 4, 4 and 4, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t4  −  2·t3  +  3·t2  −  t  +  1
    (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-4,-4,-4. 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 256

Ring generators

The cohomology ring has 14 minimal generators of maximal degree 4:

  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_3, an element of degree 2
  5. b_2_4, an element of degree 2
  6. b_2_5, an element of degree 2
  7. b_2_6, an element of degree 2
  8. a_3_7, a nilpotent element of degree 3
  9. a_3_8, a nilpotent element of degree 3
  10. b_3_12, an element of degree 3
  11. b_3_13, an element of degree 3
  12. b_4_19, an element of degree 4
  13. c_4_21, a Duflot element of degree 4
  14. c_4_22, a Duflot element of degree 4

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 256

Ring relations

There are 53 minimal relations of maximal degree 8:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_1_0·b_1_2
  4. b_2_3·a_1_0
  5. b_2_4·a_1_0
  6. b_2_4·b_1_1 + b_2_3·b_1_2
  7. b_2_5·a_1_0
  8. b_2_6·b_1_2 + b_2_5·b_1_2 + b_2_6·a_1_0
  9. b_2_6·b_1_2 + b_2_6·b_1_1 + b_2_5·b_1_2 + b_2_5·b_1_1
  10. b_2_5·b_1_22 + b_2_42
  11. b_2_5·b_1_1·b_1_2 + b_2_3·b_2_4
  12. b_2_5·b_1_12 + b_2_32
  13. b_2_52 + b_2_42 + b_2_3·b_1_22 + b_2_3·b_1_1·b_1_2 + b_2_3·b_2_4 + b_2_32
  14. b_2_5·b_2_6 + b_2_42 + b_2_3·b_1_22 + b_2_3·b_1_1·b_1_2 + b_2_3·b_2_6 + b_2_3·b_2_5
       + b_2_3·b_2_4 + b_2_32 + b_1_2·a_3_7
  15. b_2_3·b_2_6 + b_2_3·b_2_5 + a_1_0·a_3_7
  16. b_2_3·b_2_6 + b_2_3·b_2_5 + b_1_1·a_3_7
  17. b_2_5·b_2_6 + b_2_4·b_2_6 + b_2_4·b_2_5 + b_2_42 + b_2_3·b_1_22 + b_2_3·b_1_1·b_1_2
       + b_2_3·b_2_6 + b_2_3·b_2_5 + b_2_3·b_2_4 + b_2_32 + b_1_2·a_3_8
  18. b_2_4·b_2_6 + b_2_4·b_2_5 + b_2_3·b_2_6 + b_2_3·b_2_5 + a_1_0·a_3_8
  19. b_2_5·b_2_6 + b_2_4·b_2_6 + b_2_4·b_2_5 + b_2_42 + b_2_3·b_1_22 + b_2_3·b_1_1·b_1_2
       + b_2_3·b_2_6 + b_2_3·b_2_5 + b_2_3·b_2_4 + b_2_32 + b_1_1·a_3_8
  20. b_1_2·b_3_12 + b_2_4·b_2_6 + b_2_42 + b_2_3·b_2_4
  21. b_2_4·b_2_6 + b_2_4·b_2_5 + a_1_0·b_3_12
  22. b_1_1·b_3_12 + b_2_5·b_2_6 + b_2_4·b_2_6 + b_2_4·b_2_5 + b_2_42 + b_2_3·b_1_22
       + b_2_3·b_1_1·b_1_2 + b_2_3·b_2_5
  23. b_2_5·b_2_6 + b_2_4·b_2_6 + b_2_4·b_2_5 + b_2_42 + b_2_3·b_1_22 + b_2_3·b_1_1·b_1_2
       + b_2_3·b_2_6 + b_2_3·b_2_5 + b_2_3·b_2_4 + b_2_32 + a_1_0·b_3_13
  24. b_2_3·a_3_7
  25. b_2_5·a_3_7
  26. b_2_4·a_3_7 + b_2_3·a_3_8
  27. b_2_5·a_3_8
  28. b_2_4·a_3_8 + b_2_4·a_3_7
  29. b_2_3·b_3_12 + b_2_3·b_1_1·b_1_22 + b_2_3·b_1_12·b_1_2 + b_2_3·b_2_5·b_1_2
       + b_2_3·b_2_5·b_1_1 + b_2_3·b_2_4·b_1_2 + b_2_32·b_1_2 + b_2_32·b_1_1 + b_2_4·a_3_7
  30. b_2_5·b_3_12 + b_2_4·b_2_5·b_1_2 + b_2_42·b_1_2 + b_2_3·b_1_23 + b_2_3·b_1_12·b_1_2
       + b_2_3·b_2_5·b_1_2 + b_2_3·b_2_5·b_1_1 + b_2_3·b_2_4·b_1_2 + b_2_32·b_1_2
       + b_2_32·b_1_1
  31. b_2_4·b_3_12 + b_2_4·b_2_5·b_1_2 + b_2_42·b_1_2 + b_2_3·b_1_23 + b_2_3·b_1_1·b_1_22
       + b_2_3·b_2_5·b_1_2 + b_2_3·b_2_4·b_1_2 + b_2_32·b_1_2
  32. b_2_6·b_3_12 + b_2_4·b_2_5·b_1_2 + b_2_42·b_1_2 + b_2_3·b_1_23 + b_2_3·b_1_12·b_1_2
       + b_2_3·b_2_5·b_1_2 + b_2_3·b_2_5·b_1_1 + b_2_3·b_2_4·b_1_2 + b_2_32·b_1_2
       + b_2_32·b_1_1 + b_2_6·a_3_8 + b_2_6·a_3_7 + b_2_62·a_1_0 + b_2_4·a_3_7
  33. b_2_6·b_3_13 + b_2_5·b_3_13 + b_2_6·a_3_8
  34. b_1_1·b_1_2·b_3_13 + b_4_19·b_1_2 + b_2_4·b_3_13 + b_2_4·b_1_23 + b_2_4·b_2_5·b_1_2
       + b_2_3·b_1_23 + b_2_3·b_2_5·b_1_2 + b_2_32·b_1_2
  35. b_4_19·a_1_0 + b_2_4·a_3_7
  36. b_1_12·b_3_13 + b_4_19·b_1_1 + b_2_3·b_3_13 + b_2_3·b_1_23 + b_2_3·b_1_1·b_1_22
       + b_2_3·b_2_5·b_1_2 + b_2_3·b_2_5·b_1_1 + b_2_32·b_1_1
  37. a_3_72
  38. a_3_82
  39. a_3_8·b_3_12 + a_3_7·a_3_8 + b_2_6·a_1_0·a_3_8
  40. b_3_122 + b_2_3·b_2_4·b_2_5 + b_2_3·b_2_42 + b_2_32·b_2_4
  41. a_3_7·b_3_12 + a_3_7·a_3_8 + b_2_6·a_1_0·a_3_7
  42. a_3_8·b_3_13
  43. a_3_7·b_3_13 + a_3_7·a_3_8
  44. b_3_132 + b_4_19·b_1_22 + b_4_19·b_1_12 + b_2_4·b_1_2·b_3_13 + b_2_4·b_1_24
       + b_2_42·b_2_5 + b_2_43 + b_2_3·b_1_13·b_1_2 + b_2_32·b_1_1·b_1_2 + b_2_32·b_1_12
       + b_2_32·b_2_5 + b_2_33 + c_4_22·b_1_12 + c_4_21·b_1_22
  45. b_2_5·b_1_1·b_3_13 + b_2_3·b_1_1·b_3_13 + b_2_3·b_4_19 + b_2_3·b_2_4·b_1_22
       + b_2_3·b_2_4·b_2_5 + b_2_32·b_1_22 + b_2_32·b_2_5 + b_2_33
  46. b_3_12·b_3_13 + b_2_5·b_1_2·b_3_13 + b_2_5·b_4_19 + b_2_3·b_2_4·b_1_22
       + b_2_3·b_2_42 + b_2_32·b_1_1·b_1_2 + b_2_32·b_2_5 + b_2_33 + a_3_7·a_3_8
       + b_2_6·a_1_0·a_3_8
  47. b_2_5·b_1_2·b_3_13 + b_2_4·b_4_19 + b_2_3·b_1_2·b_3_13 + b_2_3·b_1_24
       + b_2_3·b_1_13·b_1_2 + b_2_3·b_2_4·b_1_22 + b_2_32·b_1_22 + b_2_32·b_1_12
       + b_2_32·b_2_5 + b_2_32·b_2_4
  48. b_3_12·b_3_13 + b_2_6·b_4_19 + b_2_5·b_1_2·b_3_13 + b_2_3·b_2_4·b_1_22
       + b_2_3·b_2_42 + b_2_32·b_1_1·b_1_2 + b_2_32·b_2_5 + b_2_33 + b_2_6·a_1_0·a_3_7
  49. b_4_19·a_3_8 + a_1_0·a_3_7·a_3_8
  50. b_4_19·a_3_7 + a_1_0·a_3_7·a_3_8
  51. b_4_19·b_3_12 + b_2_4·b_2_5·b_3_13 + b_2_42·b_3_13 + b_2_3·b_1_22·b_3_13
       + b_2_3·b_4_19·b_1_2 + b_2_3·b_2_4·b_3_13 + a_1_0·a_3_7·a_3_8
  52. b_4_19·b_3_13 + b_4_19·b_1_1·b_1_22 + b_4_19·b_1_13 + b_2_4·b_1_22·b_3_13
       + b_2_4·b_2_5·b_3_13 + b_2_43·b_1_2 + b_2_3·b_1_22·b_3_13 + b_2_3·b_1_25
       + b_2_3·b_1_14·b_1_2 + b_2_3·b_4_19·b_1_1 + b_2_3·b_2_5·b_3_13 + b_2_32·b_3_13
       + b_2_32·b_1_23 + b_2_32·b_1_1·b_1_22 + b_2_32·b_1_12·b_1_2 + b_2_32·b_1_13
       + b_2_32·b_2_4·b_1_2 + b_2_33·b_1_1 + a_1_0·a_3_7·a_3_8 + c_4_22·b_1_13
       + c_4_21·b_1_1·b_1_22 + b_2_4·c_4_21·b_1_2 + b_2_3·c_4_22·b_1_1
  53. b_4_19·b_1_12·b_1_22 + b_4_19·b_1_14 + b_4_192 + b_2_44 + b_2_3·b_1_26
       + b_2_3·b_1_15·b_1_2 + b_2_3·b_4_19·b_1_1·b_1_2 + b_2_3·b_2_5·b_4_19
       + b_2_3·b_2_4·b_1_24 + b_2_32·b_1_13·b_1_2 + b_2_32·b_1_14 + b_2_32·b_2_4·b_2_5
       + b_2_33·b_1_1·b_1_2 + b_2_33·b_1_12 + b_2_33·b_2_5 + b_2_34 + c_4_22·b_1_14
       + c_4_21·b_1_12·b_1_22 + b_2_42·c_4_21 + b_2_32·c_4_22

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 256

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • 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_4_21, a Duflot element of degree 4
    2. c_4_22, a Duflot element of degree 4
    3. b_1_22 + b_1_1·b_1_2 + b_1_12 + b_2_6 + b_2_5, an element of degree 2
    4. b_1_1·b_1_22 + b_1_12·b_1_2, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 6, 9].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
  • We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 2 elements of degree 2.

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 256

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  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_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_2_6 → 0, an element of degree 2
  8. a_3_7 → 0, an element of degree 3
  9. a_3_8 → 0, an element of degree 3
  10. b_3_12 → 0, an element of degree 3
  11. b_3_13 → 0, an element of degree 3
  12. b_4_19 → 0, an element of degree 4
  13. c_4_21 → c_1_14, an element of degree 4
  14. c_4_22 → c_1_04, an element of degree 4

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 → 0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_2_6 → c_1_22, an element of degree 2
  8. a_3_7 → 0, an element of degree 3
  9. a_3_8 → 0, an element of degree 3
  10. b_3_12 → 0, an element of degree 3
  11. b_3_13 → 0, an element of degree 3
  12. b_4_19 → 0, an element of degree 4
  13. c_4_21 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  14. c_4_22 → c_1_02·c_1_22 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_2_6 → 0, an element of degree 2
  8. a_3_7 → 0, an element of degree 3
  9. a_3_8 → 0, an element of degree 3
  10. b_3_12 → 0, an element of degree 3
  11. b_3_13 → c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  12. b_4_19 → c_1_1·c_1_2·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  13. c_4_21 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
       + c_1_12·c_1_22 + c_1_14, an element of degree 4
  14. c_4_22 → c_1_1·c_1_33 + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
       + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_23 + c_1_02·c_1_32 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_2_3 → c_1_2·c_1_3, an element of degree 2
  5. b_2_4 → c_1_32, an element of degree 2
  6. b_2_5 → c_1_32, an element of degree 2
  7. b_2_6 → c_1_32, an element of degree 2
  8. a_3_7 → 0, an element of degree 3
  9. a_3_8 → 0, an element of degree 3
  10. b_3_12 → c_1_2·c_1_32, an element of degree 3
  11. b_3_13 → c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  12. b_4_19 → c_1_23·c_1_3 + c_1_1·c_1_33 + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_2·c_1_3
       + c_1_02·c_1_22, an element of degree 4
  13. c_4_21 → c_1_34 + c_1_22·c_1_32 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3
       + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14, an element of degree 4
  14. c_4_22 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_1·c_1_2·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23
       + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_3, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_3 → c_1_32, an element of degree 2
  5. b_2_4 → c_1_2·c_1_3, an element of degree 2
  6. b_2_5 → c_1_32, an element of degree 2
  7. b_2_6 → c_1_32, an element of degree 2
  8. a_3_7 → 0, an element of degree 3
  9. a_3_8 → 0, an element of degree 3
  10. b_3_12 → c_1_2·c_1_32, an element of degree 3
  11. b_3_13 → c_1_2·c_1_32 + c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  12. b_4_19 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_23·c_1_3, an element of degree 4
  13. c_4_21 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3
       + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14, an element of degree 4
  14. c_4_22 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_1·c_1_23
       + c_1_12·c_1_22 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_22 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_3 + c_1_2, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_3 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  5. b_2_4 → c_1_2·c_1_3, an element of degree 2
  6. b_2_5 → c_1_32, an element of degree 2
  7. b_2_6 → c_1_32, an element of degree 2
  8. a_3_7 → 0, an element of degree 3
  9. a_3_8 → 0, an element of degree 3
  10. b_3_12 → 0, an element of degree 3
  11. b_3_13 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_32 + c_1_0·c_1_22 + c_1_02·c_1_3
       + c_1_02·c_1_2, an element of degree 3
  12. b_4_19 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_23
       + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22, an element of degree 4
  13. c_4_21 → c_1_22·c_1_32 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14, an element of degree 4
  14. c_4_22 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_0·c_1_2·c_1_32
       + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22
       + c_1_04, an element of degree 4

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 256


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: 11.06.2010