Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#12678 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #442 |
| commutator | C8 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1776 |
| prime radical | C23×C19 | #137 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
292 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)(20,21,24,31,26,32,40,30,27,33,41,47,25,29,23,39)(22,34,36,45,37,46,50,49,38,48,51,28,42,44,35,43);(20,22,25,37,27,38,26,42)(21,28,29,43,33,45,32,49)(23,36,41,50,40,51,24,35)(30,46,31,48,39,44,47,34);(21,29,33,32)(23,40)(24,41)(28,43,45,49)(30,47,39,31)(34,44,48,46)(35,50)(36,51)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5666);