Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#23043 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | D42 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | #2361 |
| prime radical | C19×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ G | |
820 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)(20,21,26,31)(22,32)(23,24,41,44)(25,29,33,30)(27,37)(28,39,43,40)(34,45)(35,51)(36,47)(38,48)(42,46)(49,50);(20,22,33,36,26,38,25,37)(21,27,30,32,31,47,29,48)(23,34,40,50,41,51,39,46)(24,42,28,45,44,49,43,35);(20,23)(21,28)(22,34)(24,29)(25,39)(26,41)(27,45)(30,44)(31,43)(32,49)(33,40)(35,47)(36,50)(37,46)(38,51)(42,48);(20,24,25,43,26,44,33,28)(21,23,29,39,31,41,30,40)(22,35,36,42,38,45,37,49)(27,46,32,34,47,50,48,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 16031);