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×#12714 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1781 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| prime radical | C19×#3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×C19×2−1+4 ◁ G | |
974 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)(22,32)(23,35,37,46)(24,38,40,29)(25,30)(26,31)(27,34,43,39)(28,33,45,41)(36,49)(42,50)(44,47)(48,51);(20,22,26,25)(21,27,31,43)(23,33,37,41)(24,34,40,39)(28,42,45,49)(29,32,38,30)(35,48,46,44)(36,47,50,51);(20,23,26,37)(21,28,31,45)(22,33,25,41)(24,36,40,50)(27,42,43,49)(29,44,38,48)(30,46,32,35)(34,47,39,51)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5702);