Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#13023 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
474 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)(24,25,29,35,44,53,28,32,31,37,27,34,30,36,45,49)(26,38,41,33,47,48,46,55,43,54,39,52,42,50,40,51);(20,21,22,23)(25,32)(27,28)(29,45)(30,44)(33,48)(34,36)(35,53)(37,49)(38,55)(39,46)(40,41)(42,47)(50,52)(51,54);(24,26)(25,33)(27,40)(28,41)(29,46)(30,47)(31,43)(32,48)(34,38)(35,54)(36,55)(37,52)(39,45)(42,44)(49,50)(51,53)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6011);