Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#7180 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #106 |
| commutator | C8 | C4×C8×C19 |
| Frattini, cosocle | C2×C4×C8 | C22×C19 |
| socle | C2×C19 | #1279 |
| prime radical | C23×C19 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 2 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,30,26,32,42,47,27,33,41,29,25,31,23,40)(22,34,36,45,38,50,35,46,39,48,43,44,37,49,51,28);(20,22)(21,28,31,44,33,46,32,45)(23,35)(24,43)(25,37)(26,38)(27,39)(29,34,47,49,30,48,40,50)(36,41)(42,51)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 168);