Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13140 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #481 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1926 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,26,32,24,30,27,42)(22,31,25,40,34,28,35,44)(23,36,37,47,39,51,38,43)(29,41,45,33,46,49,48,50);(20,22,27,35,24,34,26,25)(21,28,42,40,30,31,32,44)(23,33,37,49,39,50,38,41)(29,43,45,36,46,47,48,51);(20,23,26,38,24,39,27,37)(21,29,42,45,30,46,32,48)(22,33,25,41,34,50,35,49)(28,43,40,36,31,47,44,51)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6128);