Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#19591 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#3247 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
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)(20,21,25,28)(22,29,31,34)(23,27,24,33)(26,32,35,30)(36,37,41,45)(38,42,47,51)(39,44,40,49)(43,48,50,46);(20,22,23,32,25,31,24,30)(21,26,33,34,28,35,27,29)(36,38)(37,42)(39,48)(40,46)(41,47)(43,49)(44,50)(45,51);(20,21)(22,26)(23,33)(24,27)(25,28)(29,30)(31,35)(32,34)(36,38)(37,43)(39,46)(40,48)(41,47)(42,49)(44,51)(45,50);(20,23,25,24)(21,27,28,33)(22,30,31,32)(26,34,35,29)(36,39,41,40)(37,44,45,49)(38,46,47,48)(42,43,51,50)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12579);