Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13262 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #459 |
| 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 | |
462 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)(24,25,28,33,29,44,42,51,31,35,41,50,30,34,27,40)(26,36,37,47,45,52,55,49,39,53,54,48,38,46,43,32);(20,21,22,23)(24,26,31,39)(25,32,35,49)(27,37,42,54)(28,43,41,55)(29,38,30,45)(33,46,50,52)(34,47,44,48)(36,51,53,40);(24,27,30,41,31,42,29,28)(25,33,44,51,35,50,34,40)(26,37,45,55,39,54,38,43)(32,46,48,53,49,52,47,36)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6250);