Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C4 |
| direct product | C19×#13117 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #476 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1920 |
| prime radical | C23×C19 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
254 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)(20,21,23,27,26,31,25,29)(22,32,35,46,36,49,33,48)(24,38,37,47,41,45,40,30)(28,39,43,34,44,50,42,51);(21,27,31,29)(28,42,44,43)(30,38,47,45)(32,48,49,46);(20,22,23,33,26,36,25,35)(21,28,29,43,31,44,27,42)(24,39,37,51,41,50,40,34)(30,46,45,49,47,48,38,32)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6105);