Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13247 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1972 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
414 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)(22,24,30,42)(23,35)(25,44,43,29)(27,34,41,32)(28,45)(33,40)(36,39)(37,46)(38,47)(48,49)(50,51);(20,22,25,32,27,29,24,31,26,30,43,34,41,44,42,21)(23,36,45,28,39,35,49,46,40,50,38,47,51,33,37,48);(20,23,25,38,27,39,24,37,26,40,43,45,41,51,42,49)(21,28,30,35,32,46,44,50,31,47,22,33,34,48,29,36)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6235);