Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12825 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #476 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1802 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,32,27,33,26,30)(22,34,36,28,37,31,25,42)(23,38,40,49,41,51,39,47)(29,44,45,35,46,43,48,50);(20,22,24,36,27,37,26,25)(21,28,30,34,33,42,32,31)(23,35,39,44,41,50,40,43)(29,38,45,49,46,51,48,47);(20,23)(21,29)(22,35)(24,39)(25,43)(26,40)(27,41)(28,38)(30,45)(31,47)(32,48)(33,46)(34,49)(36,44)(37,50)(42,51)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5813);