Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13188 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #476 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1948 |
| prime radical | C23×C19 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
280 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)(20,21,25,32,27,33,26,31)(22,34,38,48,39,45,37,51)(23,40,41,29,42,30,24,43)(28,44,46,36,47,50,49,35);(20,22)(21,28,33,47)(23,35)(24,44)(25,37)(26,38)(27,39)(29,45,43,34)(30,48,40,51)(31,46,32,49)(36,42)(41,50);(20,23,25,41,27,42,26,24)(21,29,31,40,33,43,32,30)(22,35,37,50,39,36,38,44)(28,45,46,51,47,34,49,48)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6176);