Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72219810 |
| commutator, hypocenter | G | C1 |
| Frattini | C24×C4 | #11757162 |
| Fitting, cosocle | C28×C4 | A5 |
| socle | C25 | #14515303 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
Only the 12 normal subgroups are shown; non-normal subgroups are omitted.
Only the group's 12 normal subgroups are shown; non-normal subgroups are omitted.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,3,5,8,12)(2,6,7,15,9)(4,14,11,16,10)(17,56,18,62,40,25,31,28,45,58,55,51,57,24,34,29,27,32,52,53)(19,44,50,37)(20,63,61,43,41,38,39,30,46,64,22,49,59,48,47,42,35,36,23,21)(26,60,54,33);(2,3,5)(4,7,8)(6,11,15)(9,13,16)(10,14,12)(17,18,21)(19,24,34)(20,27,23)(22,31,46)(25,28,41)(26,39,30)(29,32,47)(33,49,59)(35,36,54)(37,52,53)(38,56,43)(40,50,62)(42,51,48)(44,45,58)(55,57,64)(60,63,61)80 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,5,8,12)(2,6,7,15,9)(4,14,11,16,10)(17,56,18,62,40,25,31,28,45,58,55,51,57,24,34,29,27,32,52,53)(19,44,50,37)(20,63,61,43,41,38,39,30,46,64,22,49,59,48,47,42,35,36,23,21)(26,60,54,33), (2,3,5)(4,7,8)(6,11,15)(9,13,16)(10,14,12)(17,18,21)(19,24,34)(20,27,23)(22,31,46)(25,28,41)(26,39,30)(29,32,47)(33,49,59)(35,36,54)(37,52,53)(38,56,43)(40,50,62)(42,51,48)(44,45,58)(55,57,64)(60,63,61));