Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#11635 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C22×C19 | C2×C22⋊C8 |
| prime radical | C19×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,26)(23,25)(24,27)(28,29,32,38,33,39,47,56,34,36,48,52,31,37,45,54)(30,40,42,57,41,50,58,55,43,53,46,59,44,35,49,51);(21,25)(22,23)(26,27)(28,30)(29,35)(31,44)(32,46)(33,41)(34,43)(36,50)(37,53)(38,55)(39,40)(42,48)(45,58)(47,49)(51,52)(54,57)(56,59);(20,22,24,23)(21,26,27,25)(29,36)(35,50)(37,39)(38,52)(40,53)(51,55)(54,56)(57,59)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4623);