Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7316 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#7317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1304 |
| commutator | C24 | C42×C19 |
| Frattini, cosocle | C23×Q8 | C22×C19 |
| prime radical | C24×C19 | C42 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C23×C19 ◁ C23×Q8×C19 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 2 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,27,25,29,36,44)(22,30,31,42,33,26,34,40)(24,37,38,43,39,28,35,45)(32,41,48,51,49,46,47,50);(20,22,24,32,25,33,39,49)(21,26,28,41,29,30,37,46)(23,34,35,48,36,31,38,47)(27,42,43,51,44,40,45,50)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 305);