Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#33362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C22×#99 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C2×D4 | C24×C19 |
| prime radical | C19×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×Q8×C19 ◁ C2×C19×2−1+4 ◁ G | |
1220 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)(22,32)(23,28)(24,29)(25,30,39,41)(26,31)(27,34,42,38)(33,43)(35,44,49,51)(36,45)(37,46)(40,47,50,48);(20,22,26,24)(21,27,31,42)(23,33,36,37)(25,38,39,34)(28,40,45,50)(29,41,32,30)(35,48,49,47)(43,44,46,51);(20,23,26,36)(21,28,31,45)(22,33,24,37)(25,35,39,49)(27,40,42,50)(29,46,32,43)(30,44,41,51)(34,47,38,48);(20,24,26,22)(21,29,31,32)(23,33,36,37)(25,38,39,34)(27,41,42,30)(28,43,45,46)(35,47,49,48)(40,44,50,51)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26350);