Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33443 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | C22×D4 |
| commutator | C4 | C23×C8×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C2×C19 | C2×C8×D4 |
| prime radical | C2×C19×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C2×C19×M32 ◁ G | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,30,24,31,39,48,26,28,41,44,40,29,45,47)(22,32,34,38,35,43,49,37,36,46,50,51,33,27,42,23);(20,22)(21,27)(23,30)(24,35)(25,34)(26,36)(28,43)(29,46)(31,32)(33,40)(37,44)(38,48)(39,49)(41,50)(42,45)(47,51);(20,23,25,32,24,38,39,43,26,37,41,46,40,51,45,27)(21,22,30,34,31,35,48,49,28,36,44,50,29,33,47,42);(21,28)(23,37)(27,43)(29,31)(30,44)(32,46)(38,51)(47,48)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26431);