Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1790 |
| commutator | C2×Q8 | C22×C4×C19 |
| Frattini, cosocle | C22×Q8 | C23×C19 |
| prime radical | C19×#3544 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C23×C19 ◁ C22×Q8×C19 ◁ G | |
730 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)(20,21,26,32,27,33,25,31)(22,34,36,50,37,48,42,44)(23,38,40,29,24,30,39,47)(28,43,45,35,46,51,49,41);(20,22)(21,28)(23,35)(24,41)(25,42)(26,36)(27,37)(29,44)(30,48)(31,45)(32,49)(33,46)(34,38)(39,43)(40,51)(47,50);(20,23,27,24)(21,29,33,47)(22,35,37,41)(25,40,26,39)(28,44,46,50)(30,32,38,31)(34,45,48,49)(36,43,42,51)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5770);