Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| Fitting | #72258201 ⋊ A5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #72209703 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | #72258201 | A5 |
| socle | C2 | #72219284 |
| LCS | G | |
| UCS | C1 ◁ C4 | |
At least 25787 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has at least 25787 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,5)(2,6)(3,9)(4,12)(7,10)(8,20)(11,41)(13,25)(14,29)(15,45)(16,32)(17,19)(18,33)(21,36)(22,38)(23,26)(24,34)(27,43)(28,42)(30,31)(35,47)(37,40)(39,51)(44,58)(46,61)(48,57)(49,60)(50,63)(52,53)(54,56)(55,59)(62,64);(2,5)(3,9)(4,12)(7,10)(8,16)(11,22)(13,25)(14,28)(17,32)(18,27)(19,34)(20,24)(21,35)(23,40)(26,37)(29,42)(30,45)(33,43)(38,41)(39,51)(44,57)(46,60)(48,61)(49,58)(50,62)(52,53)(54,56)(55,59);(1,32,43,64,60,22,6,24,42,63,57,37)(2,36,51,50,15,3,5,47,52,62,31,10)(4,23,33,56,18,11,13,41,29,59,14,26)(7,8,16,39,58,49,9,19,34,53,61,48)(12,55,25,54)(17,21,27,46,45,38,20,35,28,44,30,40)58 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((1,5)(2,6)(3,9)(4,12)(7,10)(8,20)(11,41)(13,25)(14,29)(15,45)(16,32)(17,19)(18,33)(21,36)(22,38)(23,26)(24,34)(27,43)(28,42)(30,31)(35,47)(37,40)(39,51)(44,58)(46,61)(48,57)(49,60)(50,63)(52,53)(54,56)(55,59)(62,64), (2,5)(3,9)(4,12)(7,10)(8,16)(11,22)(13,25)(14,28)(17,32)(18,27)(19,34)(20,24)(21,35)(23,40)(26,37)(29,42)(30,45)(33,43)(38,41)(39,51)(44,57)(46,60)(48,61)(49,58)(50,62)(52,53)(54,56)(55,59), (1,32,43,64,60,22,6,24,42,63,57,37)(2,36,51,50,15,3,5,47,52,62,31,10)(4,23,33,56,18,11,13,41,29,59,14,26)(7,8,16,39,58,49,9,19,34,53,61,48)(12,55,25,54)(17,21,27,46,45,38,20,35,28,44,30,40));