Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12749 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23)(24,25,30,35,31,27,29,34)(26,36,40,52,41,53,39,50)(28,43,46,42,47,51,45,33)(32,44,49,55,37,54,48,38);(24,26)(25,32)(27,37)(28,44)(29,39)(30,40)(31,41)(33,50)(34,48)(35,49)(36,43)(38,45)(42,52)(46,55)(47,54)(51,53);(21,23)(24,27,31,25)(26,37,41,32)(28,42,47,33)(29,34,30,35)(36,38,53,55)(39,48,40,49)(43,45,51,46)(44,52,54,50)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5737);