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×#12649 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #442 |
| commutator | C8 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1762 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
428 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)(22,32,35,48)(23,29)(24,40)(25,42)(26,31)(27,34,44,49)(28,39)(30,41)(33,45,37,51)(36,43,50,46)(38,47);(20,22,24,36,41,33,38,49,26,35,39,50,25,37,23,34)(21,27,29,45,42,43,28,32,31,44,47,51,30,46,40,48);(22,33,35,37)(23,38)(24,39)(27,43,44,46)(28,40)(29,47)(32,45,48,51)(34,50,49,36)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5637);