Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#11644 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C22×C19 | C2×C22⋊C8 |
| prime radical | C23×C19 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
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)(22,29)(23,27)(24,33)(25,28)(26,31)(30,34)(32,35)(36,37,39,44,41,42,47,49,40,43,48,50,38,45,46,51);(20,22,25,32)(21,26,28,34)(23,31,24,30)(27,35,33,29)(36,38,40,41)(37,42,43,45)(39,47,48,46)(44,51,50,49);(20,23,25,24)(21,27,28,33)(22,30,32,31)(26,29,34,35)(36,38,40,41)(37,42,43,45)(39,46,48,47)(44,49,50,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4632);