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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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