Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#12011 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | C2×D8 |
| commutator | C8 | C22×C8×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C2×C19 | C2×#335 |
| prime radical | C23×C19 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
220 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)(20,21,24,29,25,31,23,39,26,32,34,28,40,30,41,46)(22,33,37,44,38,48,36,27,35,43,49,45,51,47,50,42);(20,22,34,49,25,38,41,50,26,35,24,37,40,51,23,36)(21,27,39,47,30,44,29,43,32,42,46,48,31,45,28,33);(22,35)(27,42)(33,43)(36,50)(37,49)(38,51)(44,45)(47,48)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4999);