Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#11685 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #327 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C2×M16 | C23×C19 |
| socle | C2×C19 | C2×#327 |
| prime radical | C19×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C22×M16×C19 ◁ G | |
448 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)(22,31,34,48)(23,28)(24,37,39,51)(25,41)(26,30)(27,33)(29,36,46,40)(32,47,35,45)(38,42)(43,49)(44,50);(20,22)(21,27)(23,32)(24,38)(25,35)(26,34)(28,42)(29,45)(30,44)(31,37)(33,40)(36,50)(39,49)(41,43)(46,47)(48,51);(20,23,26,25)(21,28,30,41)(22,32,34,35)(24,36,39,40)(27,42,44,43)(29,37,46,51)(31,47,48,45)(33,38,50,49)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4673);