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×#12623 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #99 |
| commutator | C23 | C2×C42×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| socle | C2×C19 | C4×#99 |
| prime radical | C19×#525 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C2×C8×C19 ◁ C2×C19×#131 ◁ G | |
470 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)(20,21,25,30,26,23,28,37)(22,31,34,48,35,45,42,51)(24,29,39,46,40,36,44,50)(27,41,43,38,32,47,49,33);(20,22)(21,27)(23,32)(24,38,40,33)(25,34)(26,35)(28,42)(29,45,36,31)(30,43)(37,49)(39,47,44,41)(46,51,50,48);(20,23,25,37,26,21,28,30)(22,32,34,49,35,27,42,43)(24,36,39,50,40,29,44,46)(31,47,48,33,45,41,51,38)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5611);