Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#12997 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×#41×C19 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2432 |
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)(21,23)(24,25,29,36,28,35,44,32,31,38,27,34,30,37,45,48)(26,39,42,47,46,33,49,51,43,53,40,54,41,52,55,50);(20,21,22,23)(25,32)(27,44)(28,30)(29,45)(33,47)(34,37)(35,36)(38,48)(39,51)(40,49)(41,46)(42,55)(50,53)(52,54);(24,26)(25,33)(27,42)(28,46)(29,40)(30,41)(31,43)(32,47)(34,51)(35,53)(36,50)(37,39)(38,52)(44,55)(45,49)(48,54)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5985);