Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#12648 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #442 |
| commutator | C8 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | AGL(1,ℤ/8) |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
422 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)(20,21)(22,23)(24,25,28,33)(26,36)(27,42,29,45)(30,47,43,35)(31,39)(32,46,34,44)(37,52)(38,51)(40,55)(41,49)(48,53)(50,54);(21,23)(24,26,29,39,30,40,46,53,28,41,27,38,43,37,44,54)(25,31,34,49,35,50,45,55,33,51,32,36,47,48,42,52);(26,37,41,40)(27,29)(31,48,51,50)(32,34)(36,52,49,55)(38,54,39,53)(42,45)(44,46)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5636);