Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13187 |
| 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 | |
294 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)(26,36,40,54)(27,42)(28,32)(29,47)(30,35)(31,38,49,37)(33,44)(34,43)(39,52,41,53)(45,50,55,48)(46,51);(24,26,29,39,30,40,43,41)(25,31,34,48,35,49,47,50)(27,37,28,45,44,38,46,55)(32,36,42,53,51,54,33,52);(20,21,22,23)(24,27,43,46,30,44,29,28)(25,32,47,33,35,51,34,42)(26,37,41,55,40,38,39,45)(31,36,50,52,49,54,48,53)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6175);