Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12766 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| socle | C2×C19 | #1790 |
| prime radical | C19×#3529 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C2×C4×C19 ◁ C2×#41×C19 ◁ G | |
760 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)(20,21,24,28,25,29,40,47)(22,30,33,48,31,44,38,50)(23,34,37,46,36,27,39,45)(26,41,43,35,42,49,51,32);(20,22)(21,26)(23,35)(24,38)(25,31)(27,44)(28,43)(29,42)(30,34)(32,36)(33,40)(37,41)(39,49)(45,50)(46,48)(47,51);(22,31)(26,42)(27,34)(32,35)(33,38)(41,49)(43,51)(45,46)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5754);