Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#32915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C24×C19 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C2×C19×#117 ◁ G | |
1330 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)(21,23)(24,25)(26,27)(28,29,33,37)(30,34,39,48)(31,41,43,51)(32,44,46,36)(35,45,50,42)(38,52,53,59)(40,56,55,49)(47,57,58,54);(20,21,22,23)(28,30)(29,34)(31,38)(32,40)(33,39)(35,47)(36,49)(37,48)(41,52)(42,54)(43,53)(44,56)(45,57)(46,55)(50,58)(51,59);(28,31,33,43)(29,35,37,50)(30,38,39,53)(32,45,46,42)(34,47,48,58)(36,51,44,41)(40,57,55,54)(49,59,56,52);(25,27)(30,39)(34,48)(38,53)(40,55)(47,58)(49,56)(52,59)(54,57)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 25903);