Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#19573 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
572 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)(20,21,25,29)(22,30,32,34)(23,33,26,27)(24,35,31,28)(37,41)(38,45)(39,42)(44,49)(46,47)(48,51);(20,22,31,27,25,32,24,33)(21,26,35,34,29,23,28,30)(36,37,42,49,40,44,39,41)(38,43,45,51,46,50,47,48);(20,23)(21,27)(22,28)(24,34)(25,26)(29,33)(30,31)(32,35)(36,38,40,46)(37,43,44,50)(39,47,42,45)(41,48,49,51);(22,32)(23,26)(27,33)(30,34)(38,46)(43,50)(45,47)(48,51)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12561);