Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C23×C4 |
| direct product | C19×#19620 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C2×#127 |
| prime radical | C19×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ G | |
712 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)(20,21,24,27)(22,28,29,35)(23,31,32,26)(25,33,34,30)(36,37)(38,45)(39,43)(40,49)(41,44)(42,47)(46,50)(48,51);(20,22,23,30,24,29,32,33)(21,25,26,28,27,34,31,35);(22,29)(25,34)(28,35)(30,33)(36,38,40,47,41,48,39,46)(37,42,43,45,44,50,49,51);(36,39,41,40)(37,43,44,49)(38,46,48,47)(42,45,50,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12608);