Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11141 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #394 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C2×C19 | #1439 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
848 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)(22,31)(23,36,40,45)(24,41)(25,29)(26,30)(27,33)(28,39,44,38)(32,42,37,51)(34,43)(35,47)(46,49,50,48);(20,22,24,33,26,35,25,34)(21,27,29,31,30,43,41,47)(23,37,38,49,40,32,39,48)(28,42,36,50,44,51,45,46);(20,23)(21,28)(22,32)(24,39)(25,38)(26,40)(27,42)(29,45)(30,44)(31,46)(33,49)(34,48)(35,37)(36,41)(43,51)(47,50)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4129);