Cohomology of group number 880 of order 128

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General information on the group

  • The group has 3 minimal generators and exponent 16.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has 2 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 10 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_9, an element of degree 3
  7. a_5_13, a nilpotent element of degree 5
  8. b_5_18, an element of degree 5
  9. b_6_24, an element of degree 6
  10. c_8_37, a Duflot regular element of degree 8

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Ring relations

There are 27 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_2_4·a_1_0
  4. b_1_1·b_1_22 + a_1_0·b_1_22 + b_2_5·a_1_0
  5. b_2_5·b_1_12 + b_2_42
  6. b_2_4·b_1_22 + a_1_0·b_3_9
  7. b_1_1·b_3_9 + b_2_4·b_1_22 + b_2_4·b_2_5
  8. b_2_5·a_1_0·b_1_22 + b_2_52·a_1_0
  9. b_2_52·b_1_1 + b_2_4·b_3_9
  10. b_3_92 + b_2_5·b_1_24 + b_2_53 + b_2_5·a_1_0·b_3_9
  11. b_2_5·a_1_0·b_3_9 + a_1_0·a_5_13
  12. b_1_1·a_5_13
  13. a_1_0·b_5_18 + b_2_5·a_1_0·b_3_9
  14. b_2_4·a_5_13
  15. b_1_22·b_5_18 + b_2_5·b_1_22·b_3_9 + b_1_22·a_5_13 + b_2_5·a_5_13
  16. b_6_24·a_1_0
  17. b_6_24·b_1_1 + b_2_4·b_5_18 + b_2_42·b_2_5·b_1_1
  18. b_3_9·b_5_18 + b_2_5·b_6_24 + b_2_42·b_2_52 + b_3_9·a_5_13 + b_2_5·a_1_0·a_5_13
  19. b_6_24·b_1_22 + b_2_5·b_1_26 + b_2_53·b_1_22 + b_3_9·a_5_13 + a_1_0·b_1_24·b_3_9
       + b_2_5·a_1_0·a_5_13
  20. b_2_5·b_1_1·b_5_18 + b_2_4·b_6_24 + b_2_43·b_2_5
  21. b_6_24·b_3_9 + b_2_5·b_1_24·b_3_9 + b_2_52·b_5_18 + b_2_42·b_2_5·b_3_9
       + b_2_5·b_1_22·a_5_13 + b_2_52·a_5_13
  22. a_5_132 + b_2_52·a_1_0·a_5_13
  23. a_5_13·b_5_18 + b_2_5·b_3_9·a_5_13
  24. b_5_182 + b_1_15·b_5_18 + b_2_53·b_1_24 + b_2_55 + b_2_4·b_1_13·b_5_18
       + b_2_4·b_2_5·b_6_24 + b_2_4·b_2_54 + b_2_42·b_6_24 + b_2_42·b_2_53
       + b_2_44·b_1_12 + b_2_44·b_2_5 + c_8_37·b_1_12
  25. b_6_24·a_5_13 + b_2_5·b_1_24·a_5_13 + b_2_53·a_5_13
  26. b_6_24·b_5_18 + b_2_52·b_1_24·b_3_9 + b_2_54·b_3_9 + b_2_4·b_1_14·b_5_18
       + b_2_4·b_2_52·b_5_18 + b_2_4·b_2_53·b_3_9 + b_2_42·b_1_12·b_5_18
       + b_2_42·b_2_52·b_3_9 + b_2_43·b_2_5·b_3_9 + b_2_45·b_1_1 + b_2_5·b_1_24·a_5_13
       + b_2_53·a_5_13 + b_2_4·c_8_37·b_1_1
  27. b_6_242 + b_2_52·b_1_28 + b_2_56 + b_2_4·b_2_52·b_6_24 + b_2_4·b_2_55
       + b_2_42·b_1_13·b_5_18 + b_2_42·b_2_5·b_6_24 + b_2_42·b_2_54
       + b_2_43·b_1_1·b_5_18 + b_2_46 + b_2_53·a_1_0·a_5_13 + b_2_42·c_8_37


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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_24 + b_1_14 + b_2_5·b_1_22 + b_2_52 + b_2_42, an element of degree 4
    3. b_2_5·b_1_24 + b_2_52·b_1_22 + b_2_42·b_1_12 + b_2_42·b_2_5, 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 2.


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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  1. a_1_00, an element of degree 1
  2. b_1_10, an element of degree 1
  3. b_1_20, an element of degree 1
  4. b_2_40, an element of degree 2
  5. b_2_50, an element of degree 2
  6. b_3_90, an element of degree 3
  7. a_5_130, an element of degree 5
  8. b_5_180, an element of degree 5
  9. b_6_240, an element of degree 6
  10. c_8_37c_1_08, an element of degree 8

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

  1. a_1_00, an element of degree 1
  2. b_1_1c_1_1, an element of degree 1
  3. b_1_20, an element of degree 1
  4. b_2_4c_1_1·c_1_2, an element of degree 2
  5. b_2_5c_1_22, an element of degree 2
  6. b_3_9c_1_23, an element of degree 3
  7. a_5_130, an element of degree 5
  8. b_5_18c_1_25 + c_1_12·c_1_23 + c_1_13·c_1_22 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  9. b_6_24c_1_26 + c_1_13·c_1_23 + c_1_02·c_1_13·c_1_2 + c_1_04·c_1_1·c_1_2, an element of degree 6
  10. c_8_37c_1_14·c_1_24 + c_1_16·c_1_22 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_13·c_1_23 + 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_1·c_1_23 + 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_00, an element of degree 1
  2. b_1_10, an element of degree 1
  3. b_1_2c_1_2 + c_1_1, an element of degree 1
  4. b_2_40, an element of degree 2
  5. b_2_5c_1_12, an element of degree 2
  6. b_3_9c_1_1·c_1_22, an element of degree 3
  7. a_5_130, an element of degree 5
  8. b_5_18c_1_13·c_1_22, an element of degree 5
  9. b_6_24c_1_12·c_1_24, an element of degree 6
  10. c_8_37c_1_28 + c_1_12·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_24 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8


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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