Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#19049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#3561 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1216 |
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)(20,21,26,31)(22,32,23,34)(24,29,25,30)(27,35,28,33)(36,37,41,39)(38,44,46,51)(40,48,50,43)(42,49,45,47);(20,22,26,23)(21,27,31,28)(24,35,25,33)(29,32,30,34)(36,38,41,46)(37,42,39,45)(40,49,50,47)(43,51,48,44);(20,23,26,22)(21,28,31,27)(24,35,25,33)(29,32,30,34)(36,39,41,37)(38,45,46,42)(40,43,50,48)(44,49,51,47);(20,24,26,25)(21,29,31,30)(22,33,23,35)(27,34,28,32)(38,46)(42,45)(44,51)(47,49)703 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12037);