Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12828 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1804 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
390 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)(20,21,23,29,26,31,25,43)(22,32,35,27,36,46,33,47)(24,38,37,44,42,51,41,30)(28,40,49,39,45,50,48,34);(20,22)(21,27,31,47)(23,33)(24,39)(25,35)(26,36)(28,44,45,30)(29,32,43,46)(34,42)(37,40)(38,48,51,49)(41,50);(20,22)(21,28)(23,35)(24,40)(25,33)(26,36)(27,44)(29,49)(30,47)(31,45)(32,38)(34,41)(37,39)(42,50)(43,48)(46,51)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5816);