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A group G is said to be superperfect if it is perfect and its second integral homology is trivial. Using the library of perfect groups compiled by Holt and Pleskin, we (very partially) list below the superperfect groups of order at most 68880 together with their third integral homology. Further information on these superperfect groups can be obtained using the GAP command DisplayInformationPerfectGroup(order,number); where "order" is the order of the group and "number" is the number of the group given in the table. (Work on the table is to be continued ... )
| List of superperfect groups |
Order | Perfect Group Number | Third integral homology
120 | 1 | [ 8, 3, 5 ]
336 | 1 | [16,3]
504 | 1 | [ 2, 9, 7 ]
1320 | 1 | [ 8, 3, 5 ]
2160 | 1 | [ 16, 3, 3, 5 ]
2184 | 1 | [ 8, 3, 7 ]
2688 | 3 | [ 8, 8, 3 ]
3840 | 6 | ???
3840 | 7 | ???
4080 | 1 | [ 3, 5, 17 ]
4896 | 1 | [ 32, 9 ]
5376 | 1 | ???
5616 | 1 | [ 8, 3 ]
6048 | 1 | [ 8, 9, 5 ]
7920 | 1 | [ 8 ]
9720 | 2 | [ 8, 9, 5 ]
12144 | 1 | [ 16, 3, 11 ]
14400 | 1 | [ 8, 8, 3, 3, 5, 5 ]
15000 | 2 | [ 8, 3, 5, 25 ]
15000 | 3 | ???
15120 | 1 | [ 16, 3, 3 ]
15360 | 4,6-7: Not yet checked if these are superperfect |
15600 | 1 | [ 16, 3, 13 ]
19656 | 1 | [ 8, 3, 7, 13 ]
21504 | 5-22: Not yet checked if these are superferfect |
24360 | 1 | [ 8, 3, 5, 7 ]
29160 | 1 | ???
29160 | 3: Not yet checked if this is superperfect |
29760 | 1 | [ 64, 3, 5 ]
32256 | 1-2: Not yet checked if these are superperfect |
32736 | 1 | [ 3, 11, 31 ]
34560 | 4 | ???
37500 | 1: Not yet checked if this is superperfect |
40320 | 1 | [ 8, 16, 3, 3, 5 ]
40320 | 3 | ???
43008 | 1-25: Not yet checked if these are superperfect |
46080 | 1: Not yet checked if this is superperfect |
50616 | 1 | [ 8, 9, 19 ]
51840 | 1 | ???
57624 | 1-2: Not yet checked if these are superperfect |
58320 | 2: Not yet checked if this is superperfect |
62400 | 1 | ???
64512 | 1-4: Not yet checked if these are superperfect |
68880 | 1 | [ 16, 3, 5, 7 ]
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