Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#13200 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #127 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1955 |
| prime radical | C19×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ G | |
438 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)(20,21,25,31,26,30,42,46,27,22,28,34,24,33,41,49)(23,35,39,50,40,32,47,48,37,36,43,51,38,29,44,45);(20,22,25,34,26,33,42,49,27,21,28,31,24,30,41,46)(23,36,43,50,40,29,44,48,37,35,39,51,38,32,47,45);(20,23,24,38,27,37,26,40)(21,29,30,35,22,32,33,36)(25,39,41,44,28,43,42,47)(31,45,46,50,34,48,49,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6188);