Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12713 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1781 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| prime radical | C19×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×C19×H(5,𝔽2) ◁ G | |
986 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)(22,31)(23,35,37,28)(24,29)(25,30)(26,32,42,47)(27,36,43,38)(33,40)(34,45)(39,44)(41,46,51,48)(49,50);(20,22,25,34)(21,26,30,42)(23,32,37,47)(24,33,39,49)(27,40,43,50)(28,31,35,45)(29,41,44,51)(36,46,38,48);(20,23)(21,27)(22,32)(24,38)(25,37)(26,40)(28,29)(30,43)(31,41)(33,48)(34,47)(35,44)(36,39)(42,50)(45,51)(46,49)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5701);