Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11284 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #397 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C2×C19 | #1442 |
| prime radical | C24×C19 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
576 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)(20,21,25,29,26,30,24,39)(22,31,33,27,34,43,40,42)(23,35,38,50,37,28,36,44)(32,46,47,49,48,45,51,41);(20,22)(21,27)(23,32,37,48)(24,40)(25,33)(26,34)(28,41,35,49)(29,43)(30,42)(31,39)(36,51,38,47)(44,46,50,45);(20,23,24,36,26,37,25,38)(21,28,29,44,30,35,39,50)(22,32,33,47,34,48,40,51)(27,41,31,45,42,49,43,46)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4272);