Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#19571 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C23×C19 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
550 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)(22,30,32,33)(23,27,26,35)(24,31,34,28)(37,41)(38,44)(39,47)(43,49)(45,46)(48,51);(20,22,24,27,25,32,34,35)(21,26,28,30,29,23,31,33)(36,37,39,41,40,43,47,49)(38,42,46,48,45,50,44,51);(20,23)(21,27)(22,31)(24,33)(25,26)(28,32)(29,35)(30,34)(36,38,40,45)(37,42,43,50)(39,46,47,44)(41,48,49,51);(22,32)(23,26)(27,35)(30,33)(38,45)(42,50)(44,46)(48,51)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12559);