Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25×C4 |
| direct product | C19×#33358 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C2×#99 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C2×D4 | C24×C19 |
| socle | C2×C19 | C22×#99 |
| prime radical | C19×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C22×#41×C19 ◁ G | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
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)(22,30,31,43)(23,27)(24,36,38,28)(25,29)(26,33)(32,44,45,51)(34,47,49,41)(35,42)(37,39)(40,46)(48,50);(20,22)(21,26)(23,32)(24,37)(25,31)(27,40)(28,43)(29,39)(30,36)(33,38)(34,48)(35,45)(41,51)(42,50)(44,47)(46,49);(22,31)(26,39)(30,43)(32,45)(33,37)(40,50)(44,51)(46,48);(20,23,25,35)(21,27,29,42)(22,32,31,45)(24,34,38,49)(26,40,39,50)(28,41,36,47)(30,44,43,51)(33,46,37,48)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26346);