Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C19 · Schur multiplier (H2) = C25 |
| direct product | C8×C19×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C19 | D4 |
| commutator | C4 | C23×C8×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C2×C19×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7,8)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10833);