Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#19586 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#3212 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
584 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)(21,25)(23,26)(27,31)(28,32)(30,33)(34,35)(36,37,41,45)(38,42,47,51)(39,44,40,49)(43,48,50,46);(20,21,26,32,24,28,23,25)(22,27,33,35,29,34,30,31)(36,38)(37,42)(39,48)(40,46)(41,47)(43,49)(44,50)(45,51);(20,22)(21,27)(23,30)(24,29)(25,31)(26,33)(28,34)(32,35)(36,38)(37,43)(39,46)(40,48)(41,47)(42,49)(44,51)(45,50);(21,28)(22,29)(25,32)(30,33)(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, 12574);