Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#21483 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #120 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | C4×#120 |
| prime radical | C19×#3025 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ G | |
1328 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)(20,21,25,30,26,24,29,38)(22,27,34,41,33,32,40,47)(23,28,37,43,36,35,42,48)(31,44,46,51,45,39,49,50);(20,22)(21,27)(23,31)(24,32)(25,34)(26,33)(28,39)(29,40)(30,41)(35,44)(36,45)(37,46)(38,47)(42,49)(43,50)(48,51);(20,23)(21,28)(22,31)(24,35)(25,37)(26,36)(27,39)(29,42)(30,43)(32,44)(33,45)(34,46)(38,48)(40,49)(41,50)(47,51);(20,24,25,38,26,21,29,30)(22,32,34,47,33,27,40,41)(23,35,37,48,36,28,42,43)(31,44,46,51,45,39,49,50)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14471);