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×#12633 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #99 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| socle | C2×C19 | C4×#99 |
| prime radical | C19×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C2×C8×C19 ◁ C2×C19×#131 ◁ G | |
460 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)(22,31,34,46,35,47,41,51)(24,36,38,50,39,29,43,44)(27,40,42,49,32,45,48,33);(20,22)(21,27)(23,32)(24,33,39,49)(25,34)(26,35)(28,41)(29,31,36,47)(30,42)(37,48)(38,40,43,45)(44,46,50,51);(20,23,25,37,26,21,28,30)(22,32,34,48,35,27,41,42)(24,36,38,50,39,29,43,44)(31,45,46,33,47,40,51,49)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5621);