Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13182 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #459 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1947 |
| prime radical | C23×C19 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
280 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)(20,21,26,31,27,32,25,42)(22,33,36,49,37,43,35,47)(23,30,24,40,39,48,38,29)(28,41,44,51,46,50,45,34);(20,22,26,36,27,37,25,35)(21,28,31,44,32,46,42,45)(23,34,24,41,39,51,38,50)(29,43,30,47,40,33,48,49);(20,23,25,38,27,39,26,24)(21,29,31,30,32,40,42,48)(22,34,35,50,37,51,36,41)(28,43,44,47,46,33,45,49)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6170);