Cohomology of group number 708 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 3 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.

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

Ring generators

The cohomology ring has 11 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. a_2_0, a nilpotent element of degree 2
  5. b_2_4, an element of degree 2
  6. b_2_5, an element of degree 2
  7. b_3_9, an element of degree 3
  8. b_5_16, an element of degree 5
  9. b_5_17, an element of degree 5
  10. b_6_23, an element of degree 6
  11. 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 28 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_1_22 + b_1_1·b_1_2 + a_1_0·b_1_2
  4. a_2_0·a_1_0
  5. b_1_12·b_1_2 + b_2_5·b_1_1 + b_2_5·a_1_0 + b_2_4·a_1_0
  6. a_2_02
  7. a_1_0·b_3_9 + b_2_5·a_1_0·b_1_2 + b_2_4·a_1_0·b_1_2 + a_2_0·b_1_1·b_1_2 + a_2_0·b_2_5
  8. b_1_1·b_3_9 + b_2_4·b_1_1·b_1_2 + b_2_4·b_1_12 + b_2_5·a_1_0·b_1_2 + b_2_4·a_1_0·b_1_2
       + a_2_0·b_1_1·b_1_2 + a_2_0·b_2_5 + a_2_0·b_2_4
  9. b_2_4·b_2_5·a_1_0 + b_2_42·a_1_0
  10. a_2_0·b_3_9 + a_2_0·b_2_5·b_1_2 + a_2_0·b_2_5·b_1_1 + a_2_0·b_2_4·b_1_2
       + a_2_0·b_2_4·b_1_1
  11. b_3_92 + b_2_4·b_2_5·b_1_12 + b_2_4·b_2_52 + b_2_42·b_1_12 + b_2_42·b_2_5
       + b_2_52·a_1_0·b_1_2 + b_2_42·a_1_0·b_1_2 + a_2_0·b_2_5·b_1_12 + a_2_0·b_2_52
       + a_2_0·b_2_4·b_1_1·b_1_2 + a_2_0·b_2_4·b_2_5
  12. a_1_0·b_5_16 + b_2_42·a_1_0·b_1_2
  13. a_1_0·b_5_17 + b_2_42·a_1_0·b_1_2 + a_2_0·b_2_4·b_1_1·b_1_2 + a_2_0·b_2_4·b_2_5
  14. b_1_1·b_5_17 + b_1_1·b_5_16 + b_2_52·b_1_12 + b_2_4·b_1_14 + b_2_4·b_2_5·b_1_12
       + b_2_42·b_1_12 + a_2_0·b_2_4·b_1_12
  15. b_1_1·b_1_2·b_5_16 + b_2_5·b_5_16 + b_2_4·b_5_17 + b_2_4·b_5_16 + b_2_4·b_2_5·b_3_9
       + b_2_4·b_2_52·b_1_2 + b_2_42·b_1_13 + b_2_42·b_2_5·b_1_1 + b_2_43·b_1_1
       + a_2_0·b_2_52·b_1_2 + a_2_0·b_2_52·b_1_1 + a_2_0·b_2_4·b_2_5·b_1_2
       + a_2_0·b_2_4·b_2_5·b_1_1 + a_2_0·b_2_42·b_1_2 + a_2_0·b_2_42·b_1_1
  16. a_2_0·b_5_17 + a_2_0·b_5_16 + a_2_0·b_2_52·b_1_1 + a_2_0·b_2_4·b_1_13
       + a_2_0·b_2_4·b_2_5·b_1_1 + a_2_0·b_2_42·b_1_1
  17. b_6_23·a_1_0 + b_2_43·a_1_0
  18. b_1_1·b_1_2·b_5_16 + b_6_23·b_1_1 + b_2_4·b_1_15 + b_2_4·b_2_52·b_1_1
       + b_2_42·b_1_13 + b_2_42·b_2_5·b_1_1 + b_2_43·b_1_1 + a_2_0·b_5_16
       + a_2_0·b_2_52·b_1_1 + a_2_0·b_2_4·b_1_13 + a_2_0·b_2_4·b_2_5·b_1_2
       + a_2_0·b_2_42·b_1_1
  19. b_3_9·b_5_17 + b_3_9·b_5_16 + b_2_5·b_1_1·b_5_16 + b_2_5·b_6_23 + b_2_42·b_1_14
       + b_2_42·b_2_5·b_1_12 + b_2_43·b_1_1·b_1_2 + b_2_43·b_1_12 + b_2_43·b_2_5
       + b_2_53·a_1_0·b_1_2 + a_2_0·b_1_2·b_5_16 + a_2_0·b_2_53 + a_2_0·b_2_4·b_2_52
       + a_2_0·b_2_42·b_1_1·b_1_2 + a_2_0·b_2_43
  20. b_3_9·b_5_16 + b_2_4·b_1_1·b_5_16 + b_2_4·b_6_23 + b_2_4·b_2_5·b_1_2·b_3_9
       + b_2_42·b_1_14 + b_2_42·b_2_52 + b_2_43·b_1_12 + b_2_43·b_2_5 + b_2_44
       + b_2_43·a_1_0·b_1_2 + a_2_0·b_2_4·b_2_52 + a_2_0·b_2_42·b_1_1·b_1_2
       + a_2_0·b_2_42·b_1_12 + a_2_0·b_2_42·b_2_5 + a_2_0·b_2_43
  21. a_2_0·b_1_2·b_5_16 + a_2_0·b_6_23 + a_2_0·b_2_4·b_1_14 + a_2_0·b_2_4·b_2_52
       + a_2_0·b_2_42·b_1_12 + a_2_0·b_2_42·b_2_5 + a_2_0·b_2_43
  22. b_6_23·b_3_9 + b_2_4·b_2_5·b_5_17 + b_2_4·b_2_5·b_5_16 + b_2_4·b_2_53·b_1_1
       + b_2_42·b_5_17 + b_2_42·b_5_16 + b_2_42·b_1_15 + b_2_43·b_3_9
       + b_2_43·b_2_5·b_1_1 + b_2_44·b_1_1 + a_2_0·b_2_4·b_5_16 + a_2_0·b_2_42·b_2_5·b_1_1
       + a_2_0·b_2_43·b_1_1
  23. b_5_172 + b_5_162 + b_2_54·b_1_12 + b_2_42·b_1_16 + b_2_42·b_2_53
       + b_2_43·b_2_5·b_1_12 + b_2_43·b_2_52 + b_2_44·b_1_12
  24. b_5_16·b_5_17 + b_5_162 + b_2_52·b_1_1·b_5_16 + b_2_4·b_1_13·b_5_16
       + b_2_4·b_2_5·b_1_2·b_5_17 + b_2_4·b_2_5·b_1_1·b_5_16 + b_2_4·b_2_5·b_6_23
       + b_2_4·b_2_53·b_1_12 + b_2_42·b_1_2·b_5_17 + b_2_42·b_1_2·b_5_16
       + b_2_42·b_1_1·b_5_16 + b_2_42·b_2_5·b_1_2·b_3_9 + b_2_42·b_2_52·b_1_12
       + b_2_43·b_2_52 + a_2_0·b_2_4·b_1_1·b_5_16 + a_2_0·b_2_4·b_6_23
       + a_2_0·b_2_4·b_2_53 + a_2_0·b_2_42·b_1_14 + a_2_0·b_2_43·b_1_12
       + a_2_0·b_2_44
  25. b_5_162 + b_1_15·b_5_16 + b_1_110 + b_2_4·b_1_13·b_5_16 + b_2_4·b_2_5·b_1_1·b_5_16
       + b_2_42·b_1_1·b_5_16 + b_2_42·b_2_52·b_1_12 + b_2_43·b_1_14 + b_2_43·b_2_52
       + b_2_45 + b_2_44·a_1_0·b_1_2 + a_2_0·b_1_13·b_5_16 + a_2_0·b_1_18
       + a_2_0·b_2_4·b_1_1·b_5_16 + a_2_0·b_2_4·b_1_16 + a_2_0·b_2_4·b_2_53
       + a_2_0·b_2_43·b_1_1·b_1_2 + a_2_0·b_2_43·b_1_12 + a_2_0·b_2_43·b_2_5
       + c_8_37·b_1_12
  26. b_6_23·b_5_17 + b_6_23·b_5_16 + b_2_53·b_5_16 + b_2_4·b_2_52·b_5_17
       + b_2_4·b_2_52·b_5_16 + b_2_4·b_2_53·b_3_9 + b_2_4·b_2_54·b_1_2 + b_2_42·b_1_17
       + b_2_42·b_2_5·b_5_16 + b_2_42·b_2_52·b_3_9 + b_2_42·b_2_53·b_1_1
       + b_2_43·b_2_52·b_1_2 + b_2_43·b_2_52·b_1_1 + b_2_45·b_1_1 + a_2_0·b_2_52·b_5_16
       + a_2_0·b_2_54·b_1_2 + a_2_0·b_2_4·b_1_12·b_5_16 + a_2_0·b_2_42·b_5_16
       + a_2_0·b_2_43·b_1_13
  27. b_6_23·b_5_16 + b_2_53·b_5_16 + b_2_55·b_1_1 + b_2_4·b_1_14·b_5_16
       + b_2_4·b_2_5·b_6_23·b_1_2 + b_2_4·b_2_52·b_5_17 + b_2_4·b_2_52·b_5_16
       + b_2_4·b_2_53·b_3_9 + b_2_4·b_2_54·b_1_2 + b_2_42·b_1_12·b_5_16
       + b_2_42·b_2_5·b_5_17 + b_2_42·b_2_5·b_5_16 + b_2_43·b_5_17 + b_2_43·b_2_5·b_3_9
       + b_2_44·b_3_9 + b_2_44·b_1_13 + b_2_44·b_2_5·b_1_2 + b_2_55·a_1_0 + b_2_45·a_1_0
       + a_2_0·b_1_14·b_5_16 + a_2_0·b_1_19 + a_2_0·b_2_54·b_1_2
       + a_2_0·b_2_4·b_2_53·b_1_1 + a_2_0·b_2_42·b_2_52·b_1_1 + a_2_0·b_2_43·b_1_13
       + a_2_0·b_2_43·b_2_5·b_1_2 + a_2_0·b_2_43·b_2_5·b_1_1 + a_2_0·b_2_44·b_1_2
       + a_2_0·b_2_44·b_1_1 + b_2_5·c_8_37·b_1_1 + b_2_5·c_8_37·a_1_0 + b_2_4·c_8_37·a_1_0
       + a_2_0·c_8_37·b_1_1
  28. b_6_232 + b_2_53·b_1_1·b_5_16 + b_2_55·b_1_12 + b_2_42·b_1_18
       + b_2_42·b_2_5·b_1_1·b_5_16 + b_2_43·b_2_52·b_1_12 + b_2_43·b_2_53
       + b_2_44·b_1_14 + b_2_44·b_2_5·b_1_12 + b_2_45·b_2_5 + b_2_46
       + a_2_0·b_2_52·b_6_23 + a_2_0·b_2_54·b_1_12 + a_2_0·b_2_4·b_2_5·b_6_23
       + a_2_0·b_2_44·b_1_1·b_1_2 + b_2_5·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 12.
  • 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_2_5·b_1_12 + b_2_52 + b_2_4·b_1_1·b_1_2 + b_2_4·b_2_5 + b_2_42, an element of degree 4
    3. b_3_9 + b_2_4·b_1_2, 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].
  • We found that there exists some filter regular HSOP formed by the first term 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 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. a_2_0 → 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_3_9 → 0, an element of degree 3
  8. b_5_16 → 0, an element of degree 5
  9. b_5_17 → 0, an element of degree 5
  10. b_6_23 → 0, an element of degree 6
  11. 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 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. a_2_0 → 0, an element of degree 2
  5. b_2_4 → c_1_22 + c_1_12, an element of degree 2
  6. b_2_5 → c_1_22, an element of degree 2
  7. b_3_9 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_5_16 → c_1_12·c_1_23 + c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_15, an element of degree 5
  9. b_5_17 → c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_15, an element of degree 5
  10. b_6_23 → c_1_26 + c_1_13·c_1_23 + c_1_15·c_1_2 + c_1_16, an element of degree 6
  11. c_8_37 → c_1_28 + c_1_12·c_1_26 + 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_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 → c_1_2, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. a_2_0 → 0, an element of degree 2
  5. b_2_4 → c_1_1·c_1_2 + c_1_12, an element of degree 2
  6. b_2_5 → c_1_22, an element of degree 2
  7. b_3_9 → 0, an element of degree 3
  8. b_5_16 → c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_13·c_1_22 + c_1_15
       + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  9. b_5_17 → c_1_1·c_1_24 + c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_15 + c_1_02·c_1_23
       + c_1_04·c_1_2, an element of degree 5
  10. b_6_23 → c_1_26 + c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_16
       + c_1_02·c_1_24 + c_1_04·c_1_22, an element of degree 6
  11. c_8_37 → c_1_28 + c_1_1·c_1_27 + 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 → c_1_2, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. a_2_0 → 0, an element of degree 2
  5. b_2_4 → c_1_1·c_1_2 + c_1_12, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_3_9 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_5_16 → c_1_25 + c_1_12·c_1_23 + c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_15
       + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  9. b_5_17 → c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_13·c_1_22 + c_1_15
       + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  10. b_6_23 → c_1_1·c_1_25 + c_1_13·c_1_23 + c_1_15·c_1_2 + c_1_16, an element of degree 6
  11. c_8_37 → c_1_28 + c_1_1·c_1_27 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24
       + c_1_16·c_1_22 + c_1_17·c_1_2 + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25
       + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_1·c_1_23 + 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