Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | D8×#126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#126 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4,5,6,7,8);(2,8)(3,7)(4,6);(10,14)(11,18)(16,23)(17,21)(19,22)(20,24);(9,10,15,21)(11,16,22,23)(12,14,13,17)(18,24,19,20);(9,11,12,19)(10,16,14,20)(13,18,15,22)(17,24,21,23)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486772);