Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#23652 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | C2×#397 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,30,26,31,24,29)(22,32,34,27,36,43,35,44)(23,37,39,45,40,46,41,28)(33,48,50,51,38,42,49,47);(20,22)(21,27)(23,38)(24,34)(25,35)(26,36)(28,42)(29,43)(30,32)(31,44)(33,40)(37,51)(39,50)(41,49)(45,48)(46,47);(20,23,24,39,26,40,25,41)(21,28,29,37,31,45,30,46)(22,33,35,50,36,38,34,49)(27,42,32,47,44,48,43,51);(20,24,26,25)(21,29,31,30)(22,34,36,35)(23,39,40,41)(27,43,44,32)(28,37,45,46)(33,49,38,50)(42,51,48,47)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 16640);