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×#13273 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #476 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25,29,33,28,45,44,55,31,36,43,53,30,35,27,34)(26,37,41,32,47,50,46,49,42,48,38,52,40,54,39,51);(20,21,22,23)(24,26,28,40,31,42,30,47)(25,32,35,49,36,52,45,51)(27,38,29,46,44,41,43,39)(33,48,34,54,53,37,55,50);(24,27,30,43,31,44,28,29)(25,33,45,55,36,53,35,34)(26,38,47,39,42,41,40,46)(32,48,51,50,52,37,49,54)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6261);