Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#33656 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C2×C19 | C2×#481 |
| prime radical | C19×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C22×M16×C19 ◁ G | |
556 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)(22,31)(23,28)(24,36)(25,40)(26,30)(27,34)(29,39)(32,46)(33,47)(35,45)(37,51)(38,44)(41,48)(42,49)(43,50);(24,37)(29,45)(31,47)(34,50)(35,39)(36,51)(38,49)(46,48);(20,22,25,41,26,33,23,32)(21,27,28,42,30,43,40,44)(24,38,39,50,37,49,35,34)(29,46,36,47,45,48,51,31);(20,23,26,25)(21,28,30,40)(22,32,33,41)(24,35,37,39)(27,42,43,44)(29,36,45,51)(31,46,47,48)(34,49,50,38)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26644);