Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#32636 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C22×D4×C19 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C2×C4×Q8×C19 ◁ G | |
674 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)(20,21,25,26)(22,24,27,23)(29,32)(30,34)(36,37)(38,41)(39,42)(40,43)(44,51)(45,49)(46,48)(47,50);(20,22,25,27)(21,23,26,24)(28,29,31,32)(30,33,34,35)(36,38)(37,41)(39,45)(40,44)(42,49)(43,51)(46,47)(48,50);(20,23,25,24)(21,22,26,27)(28,30,31,34)(29,33,32,35)(37,42)(38,44)(40,47)(41,48)(45,46)(49,51);(20,24,25,23)(21,22,26,27)(29,32)(33,35)(37,42)(41,49)(43,50)(48,51)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 25624);