Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#33637 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C2×C19 | C2×#476 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 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)(22,34)(23,38,41,51)(24,30)(25,32)(26,42)(27,33)(28,36)(29,35,45,50)(31,37)(39,48,40,46)(43,47,49,44);(20,22,25,36,27,37,24,26)(21,28,30,34,33,42,32,31)(23,35,40,49,41,50,39,43)(29,38,44,48,45,51,47,46);(20,23)(21,29)(22,35)(24,39)(25,40)(26,43)(27,41)(28,38)(30,44)(31,46)(32,47)(33,45)(34,48)(36,49)(37,50)(42,51);(20,24,27,25)(21,30,33,32)(22,26,37,36)(23,39,41,40)(28,34,42,31)(29,44,45,47)(35,43,50,49)(38,48,51,46)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26625);