Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#32737 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C2×C19×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C2×C19×#117 ◁ G | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,26)(22,25,27,24)(28,29)(30,32)(31,33)(34,35)(36,37,40,44)(38,45,47,42)(39,48,41,43)(46,51,49,50);(20,22)(21,24)(23,27)(25,26)(28,30)(29,32)(31,34)(33,35)(38,46)(39,41)(42,50)(43,48)(45,51)(47,49);(20,22)(21,25)(23,27)(24,26)(29,33)(32,35)(36,38,40,47)(37,42,44,45)(39,46,41,49)(43,51,48,50);(22,27)(24,25)(30,34)(32,35)(36,39,40,41)(37,43,44,48)(38,46,47,49)(42,51,45,50)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 25725);