Cohomology of group number 513 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 4.
  • It is non-abelian.
  • It has p-Rank 5.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 5.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 5 and depth 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    ( − 1) · (t2  +  1)
    (t  +  1)2 · (t  −  1)5
  • The a-invariants are -∞,-∞,-∞,-5,-5,-5. 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 15 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. b_2_7, an element of degree 2
  9. c_2_8, a Duflot regular element of degree 2
  10. b_3_16, an element of degree 3
  11. b_3_17, an element of degree 3
  12. b_3_18, an element of degree 3
  13. b_4_28, an element of degree 4
  14. b_4_30, an element of degree 4
  15. c_4_37, a Duflot regular element of degree 4

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

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_3·b_1_1
  6. b_2_4·a_1_0
  7. b_2_5·b_1_1 + b_2_6·a_1_0
  8. b_2_6·b_1_1 + b_2_5·b_1_1
  9. b_2_5·b_1_1 + b_2_7·a_1_0
  10. b_2_5·b_1_22 + b_2_32
  11. b_2_6·b_1_22 + b_2_3·b_2_4
  12. b_2_4·b_2_5 + b_2_3·b_2_6
  13. b_2_7·b_1_22 + b_2_4·b_1_1·b_1_2 + b_2_42 + c_2_8·b_1_12
  14. b_2_4·b_2_6 + b_2_3·b_2_7
  15. b_2_62 + b_2_5·b_2_7
  16. b_1_2·b_3_16 + b_2_4·b_2_6 + b_2_4·b_2_5
  17. a_1_0·b_3_16
  18. b_1_1·b_3_16
  19. a_1_0·b_3_17
  20. a_1_0·b_3_18
  21. b_1_2·b_3_17 + b_1_1·b_3_18 + b_2_4·b_2_7 + b_2_4·b_2_6
  22. b_2_5·b_2_7·b_1_2 + b_2_5·b_2_6·b_1_2 + b_2_3·b_3_16
  23. b_2_6·b_2_7·b_1_2 + b_2_5·b_2_7·b_1_2 + b_2_4·b_3_16
  24. b_2_6·b_2_7·b_1_2 + b_2_5·b_2_7·b_1_2 + b_2_3·b_3_17
  25. b_2_6·b_3_16 + b_2_5·b_3_17 + b_2_5·b_2_6·a_1_0
  26. b_2_7·b_3_16 + b_2_6·b_3_17 + b_2_5·b_2_6·a_1_0
  27. b_4_28·b_1_2 + b_2_3·b_3_18 + b_2_3·b_2_7·b_1_2
  28. b_4_28·a_1_0
  29. b_4_28·b_1_1
  30. b_4_30·b_1_2 + b_2_4·b_3_18 + b_2_4·b_2_7·b_1_2 + b_2_3·b_2_7·b_1_2 + b_2_7·c_2_8·b_1_1
       + b_2_6·c_2_8·a_1_0
  31. b_4_30·a_1_0
  32. b_4_30·b_1_1 + b_2_72·b_1_2 + b_2_6·b_2_7·b_1_2 + b_2_4·b_3_17
  33. b_3_162 + b_2_5·b_2_72 + b_2_52·b_2_7
  34. b_3_16·b_3_17 + b_2_6·b_2_72 + b_2_5·b_2_6·b_2_7
  35. b_3_182 + b_2_7·b_1_2·b_3_18 + b_2_6·b_1_2·b_3_18 + b_2_32·b_2_6 + c_4_37·b_1_22
       + b_2_72·c_2_8 + b_2_5·b_2_7·c_2_8
  36. b_3_172 + b_2_7·b_1_1·b_3_17 + b_2_73 + b_2_5·b_2_72 + c_4_37·b_1_12
  37. b_2_5·b_1_2·b_3_18 + b_2_3·b_4_28 + b_2_32·b_2_7
  38. b_2_6·b_1_2·b_3_18 + b_2_4·b_4_28 + b_2_3·b_2_4·b_2_7
  39. b_3_16·b_3_18 + b_2_7·b_4_28 + b_2_6·b_4_28 + b_2_3·b_2_72 + b_2_3·b_2_6·b_2_7
  40. b_2_6·b_1_2·b_3_18 + b_2_3·b_4_30 + b_2_3·b_2_4·b_2_7 + b_2_32·b_2_7
  41. b_3_16·b_3_18 + b_2_7·b_4_28 + b_2_5·b_4_30 + b_2_3·b_2_72 + b_2_3·b_2_6·b_2_7
       + b_2_3·b_2_5·b_2_7
  42. b_2_7·b_1_2·b_3_18 + b_2_4·b_1_1·b_3_18 + b_2_4·b_4_30 + b_2_42·b_2_7
       + b_2_3·b_2_4·b_2_7 + c_2_8·b_1_1·b_3_17
  43. b_3_17·b_3_18 + b_2_7·b_1_1·b_3_18 + b_2_7·b_4_30 + b_2_7·b_4_28 + b_2_4·b_2_72
       + c_4_37·b_1_1·b_1_2
  44. b_2_7·b_4_28 + b_2_6·b_4_30 + b_2_3·b_2_6·b_2_7
  45. b_4_28·b_3_17 + b_2_6·b_2_7·b_3_18 + b_2_5·b_2_7·b_3_18 + b_2_3·b_2_7·b_3_17
  46. b_4_28·b_3_18 + b_2_3·b_2_6·b_3_18 + b_2_3·b_2_5·b_2_6·b_1_2 + b_2_6·c_2_8·b_3_17
       + b_2_5·c_2_8·b_3_17 + b_2_3·c_4_37·b_1_2
  47. b_4_28·b_3_16 + b_2_5·b_2_7·b_3_18 + b_2_5·b_2_6·b_3_18 + b_2_3·b_2_6·b_3_17
  48. b_4_30·b_3_17 + b_2_72·b_3_18 + b_2_6·b_2_7·b_3_18 + b_2_4·b_2_7·b_3_17
       + b_2_3·b_2_7·b_3_17 + b_2_4·c_4_37·b_1_1
  49. b_4_30·b_3_18 + b_2_3·b_2_5·b_2_6·b_1_2 + b_2_32·b_3_16 + b_2_7·c_2_8·b_3_17
       + b_2_6·c_2_8·b_3_17 + b_2_4·c_4_37·b_1_2
  50. b_4_30·b_3_16 + b_2_6·b_2_7·b_3_18 + b_2_5·b_2_7·b_3_18 + b_2_3·b_2_7·b_3_17
       + b_2_3·b_2_6·b_3_17
  51. b_4_282 + b_2_3·b_2_6·b_4_30 + b_2_3·b_2_5·b_4_30 + b_2_32·b_2_5·b_2_7
       + b_2_32·b_2_5·b_2_6 + b_2_5·b_2_72·c_2_8 + b_2_52·b_2_7·c_2_8 + b_2_32·c_4_37
  52. b_4_28·b_4_30 + b_2_3·b_2_7·b_4_30 + b_2_32·b_2_5·b_2_7 + b_2_6·b_2_72·c_2_8
       + b_2_5·b_2_6·b_2_7·c_2_8 + b_2_3·b_2_4·c_4_37
  53. b_4_302 + b_2_4·b_2_7·b_4_30 + b_2_3·b_2_7·b_4_30 + b_2_32·b_2_6·b_2_7
       + b_2_73·c_2_8 + b_2_5·b_2_72·c_2_8 + b_2_42·c_4_37

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 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_2_8, a Duflot regular element of degree 2
    2. c_4_37, a Duflot regular element of degree 4
    3. b_1_24 + b_1_12·b_1_22 + b_1_14 + b_2_7·b_1_1·b_1_2 + b_2_72 + b_2_5·b_2_7
         + b_2_52 + b_2_4·b_1_1·b_1_2 + b_2_42 + b_2_3·b_2_7 + b_2_3·b_2_6 + b_2_3·b_2_4
         + b_2_32 + c_2_8·b_1_12, an element of degree 4
    4. b_1_12·b_1_24 + b_1_14·b_1_22 + b_2_7·b_1_13·b_1_2 + b_2_72·b_1_1·b_1_2
         + b_2_72·b_1_12 + b_2_5·b_2_72 + b_2_52·b_2_7 + b_2_4·b_1_1·b_1_23
         + b_2_4·b_2_7·b_1_1·b_1_2 + b_2_42·b_1_22 + b_2_42·b_2_7 + b_2_3·b_2_72
         + b_2_3·b_2_5·b_2_6 + b_2_3·b_2_4·b_1_22 + b_2_32·b_1_22 + b_2_32·b_2_7
         + b_2_32·b_2_5 + c_2_8·b_1_12·b_1_22 + b_2_7·c_2_8·b_1_12, an element of degree 6
    5. b_2_72·b_1_12·b_1_2 + b_2_4·b_1_12·b_1_23 + b_2_4·b_1_14·b_1_2
         + b_2_4·b_2_7·b_1_12·b_1_2 + b_2_42·b_1_1·b_1_22 + b_2_42·b_1_13
         + b_2_42·b_2_7·b_1_1 + b_2_3·b_2_6·b_3_17 + b_2_3·b_2_5·b_3_17 + b_2_3·b_2_4·b_3_17
         + b_2_3·b_2_42·b_1_2 + b_2_32·b_3_17 + b_2_32·b_3_16 + b_2_32·b_2_4·b_1_2
         + c_2_8·b_1_13·b_1_22 + c_2_8·b_1_15 + b_2_7·c_2_8·b_1_13, an element of degree 7
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 11, 18].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].
  • We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 3 elements of degree 2.

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 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. b_2_7 → 0, an element of degree 2
  9. c_2_8 → c_1_02, an element of degree 2
  10. b_3_16 → 0, an element of degree 3
  11. b_3_17 → 0, an element of degree 3
  12. b_3_18 → 0, an element of degree 3
  13. b_4_28 → 0, an element of degree 4
  14. b_4_30 → 0, an element of degree 4
  15. c_4_37 → c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

  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 → c_1_3·c_1_4 + c_1_0·c_1_2, 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. b_2_7 → c_1_42 + c_1_2·c_1_4, an element of degree 2
  9. c_2_8 → c_1_0·c_1_3 + c_1_02, an element of degree 2
  10. b_3_16 → 0, an element of degree 3
  11. b_3_17 → c_1_43 + c_1_22·c_1_4 + c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  12. b_3_18 → c_1_3·c_1_42 + c_1_2·c_1_3·c_1_4 + c_1_1·c_1_2·c_1_3 + c_1_12·c_1_3 + c_1_0·c_1_42
       + c_1_0·c_1_2·c_1_4, an element of degree 3
  13. b_4_28 → 0, an element of degree 4
  14. b_4_30 → c_1_1·c_1_2·c_1_3·c_1_4 + c_1_12·c_1_3·c_1_4 + c_1_0·c_1_43 + c_1_0·c_1_22·c_1_4
       + c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2, an element of degree 4
  15. c_4_37 → c_1_1·c_1_2·c_1_42 + c_1_1·c_1_22·c_1_4 + c_1_12·c_1_42 + c_1_12·c_1_2·c_1_4
       + c_1_12·c_1_22 + c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

  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_3 → c_1_2·c_1_4, 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_42, an element of degree 2
  7. b_2_6 → c_1_3·c_1_4, an element of degree 2
  8. b_2_7 → c_1_32, an element of degree 2
  9. c_2_8 → c_1_0·c_1_2 + c_1_02, an element of degree 2
  10. b_3_16 → c_1_3·c_1_42 + c_1_32·c_1_4, an element of degree 3
  11. b_3_17 → c_1_32·c_1_4 + c_1_33, an element of degree 3
  12. b_3_18 → c_1_2·c_1_32 + c_1_1·c_1_2·c_1_4 + c_1_12·c_1_2 + c_1_0·c_1_3·c_1_4 + c_1_0·c_1_32, an element of degree 3
  13. b_4_28 → c_1_1·c_1_2·c_1_42 + c_1_12·c_1_2·c_1_4 + c_1_0·c_1_3·c_1_42
       + c_1_0·c_1_32·c_1_4, an element of degree 4
  14. b_4_30 → c_1_2·c_1_32·c_1_4 + c_1_1·c_1_2·c_1_3·c_1_4 + c_1_12·c_1_2·c_1_3
       + c_1_0·c_1_32·c_1_4 + c_1_0·c_1_33, an element of degree 4
  15. c_4_37 → c_1_3·c_1_43 + c_1_33·c_1_4 + c_1_1·c_1_3·c_1_42 + c_1_1·c_1_32·c_1_4
       + c_1_12·c_1_42 + c_1_12·c_1_3·c_1_4 + c_1_12·c_1_32 + c_1_14, an element of degree 4

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