Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11090 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #393 |
| commutator | C24 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C2×C19 | #1439 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
852 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)(20,21,24,29,25,28,38,43)(22,26,32,41,31,40,47,50)(23,33,36,44,35,27,37,42)(30,45,46,51,34,39,48,49);(20,22)(21,26)(23,34,35,30)(24,32)(25,31)(27,39,33,45)(28,40)(29,41)(36,48,37,46)(38,47)(42,49,44,51)(43,50);(20,23)(21,27)(22,30)(24,37)(25,35)(26,39)(28,33)(29,44)(31,34)(32,48)(36,38)(40,45)(41,51)(42,43)(46,47)(49,50)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4078);