Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12725 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12726 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1782 |
| commutator | C24 | C22×C4×C19 |
| Frattini, cosocle | C22×D4 | C23×C19 |
| prime radical | C19×#534 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C23×C19 ◁ C19×#518 ◁ G | |
708 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)(22,30)(23,34)(24,37,38,28)(25,29)(26,32,40,47)(27,36)(31,44)(33,43)(35,41,50,49)(39,51,48,45)(42,46);(20,22,25,33)(21,26,29,40)(23,31,36,46)(24,32,38,47)(27,39,34,48)(28,30,37,43)(35,45,50,51)(41,42,49,44);(20,23)(21,27,29,34)(22,31)(24,35)(25,36)(26,39,40,48)(28,41,37,49)(30,42,43,44)(32,45)(33,46)(38,50)(47,51)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5714);