Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13271 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #476 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1987 |
| prime radical | C23×C19 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
332 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)(20,21,24,30,26,31,40,50,27,32,39,29,25,42,23,38)(22,33,34,45,36,49,51,28,37,46,35,47,43,44,41,48);(20,22)(21,28,32,48)(23,34)(24,41)(25,36)(26,43)(27,37)(29,44,30,49)(31,45,42,47)(33,38,46,50)(35,40)(39,51);(20,23,25,39,27,40,26,24)(21,29,31,38,32,30,42,50)(22,34,36,51,37,35,43,41)(28,44,45,46,48,49,47,33)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6259);