Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13291 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13291 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1993 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
306 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)(22,33,37,49)(23,39,42,47)(24,43)(25,30)(26,32)(27,28,41,46)(29,34)(31,45)(35,44)(36,40,38,51)(48,50);(20,22,34,47,45,36,24,28,26,37,50,39,25,38,44,46)(21,27,35,51,30,42,48,49,32,41,43,40,31,23,29,33);(20,23,26,42)(21,28)(22,35)(24,40,44,51)(25,41,45,27)(29,38)(30,39)(31,47)(32,46)(33,34,49,50)(36,48)(37,43)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6279);