Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C23×C4 |
| direct product | C19×#33701 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C22×C19 | C22×D8 |
| prime radical | C19×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 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)(22,25)(23,24)(26,27)(28,29)(30,43)(31,36)(32,38)(33,52)(34,54)(35,42)(37,46)(39,50)(40,53)(41,51)(44,56)(45,57)(47,55)(48,59)(49,58);(21,24)(25,27)(28,30,34,45,32,44,33,31,35,46,50,47,51,48,53,49)(29,36,40,56,41,57,39,43,42,58,52,59,38,55,54,37);(20,22,23,26)(21,25,24,27)(28,31,33,48,51,45,34,46,35,49,53,44,32,47,50,30)(29,37,39,57,38,59,40,36,42,43,54,55,41,56,52,58);(28,32,35,51)(29,38,42,41)(30,44,46,48)(31,47,49,45)(33,50,53,34)(36,55,58,57)(37,59,43,56)(39,40,54,52)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26689);