Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11251 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #396 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C2×C19 | #1441 |
| prime radical | C24×C19 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
620 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)(22,32)(23,37,40,45)(24,30)(25,29)(26,31)(27,34)(28,39,46,41)(33,49,51,42)(35,43)(36,47)(38,44,50,48);(20,22,24,34,26,36,25,35)(21,27,29,32,31,43,30,47)(23,38,39,33,40,50,41,51)(28,44,45,42,46,48,37,49);(20,23,24,39,26,40,25,41)(21,28,29,45,31,46,30,37)(22,33,34,50,36,51,35,38)(27,42,32,48,43,49,47,44)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4239);