Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12748 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23)(24,25,28,34,30,35,29,46)(26,36,38,51,39,50,45,32)(27,40,42,52,44,54,43,33)(31,41,47,53,49,55,48,37);(24,26)(25,31)(27,41)(28,38)(29,45)(30,39)(32,33)(34,47)(35,49)(36,40)(37,43)(42,53)(44,55)(46,48)(50,54)(51,52);(21,23)(24,26)(25,32)(27,37)(28,45)(29,38)(30,39)(31,33)(34,50)(35,51)(36,46)(40,48)(41,43)(42,55)(44,53)(47,54)(49,52)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5736);