Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33468 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #442 |
| commutator | C8 | C23×C4×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C2×C19 | C2×#442 |
| prime radical | C19×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
652 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)(22,33,36,43)(23,37)(24,30)(25,29)(26,41)(27,32)(28,34,45,51)(31,39)(35,48,42,49)(38,47)(40,44,50,46);(20,22)(21,28)(23,34)(24,35)(25,40)(26,42)(27,36)(29,43)(30,44)(31,48)(32,45)(33,47)(37,49)(38,50)(39,51)(41,46);(20,23,26,25,27,39,24,38)(21,29,30,37,32,47,41,31)(22,34,42,40,36,51,35,50)(28,43,44,49,45,33,46,48);(20,24,27,26)(21,30,32,41)(22,35,36,42)(23,38,39,25)(28,44,45,46)(29,37,47,31)(33,48,43,49)(34,50,51,40)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26456);