Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#23049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4,6,5)(7,8,13,18,14,10,12,17)(9,19,16,15,22,11,21,20);(3,5,6,4)(7,9)(8,15)(10,20)(11,17)(12,16)(13,21)(14,22)(18,19);(7,10)(8,14)(9,20)(11,21)(12,18)(13,17)(15,22)(16,19);(7,11,13,20,14,19,12,15)(8,16,18,22,10,21,17,9)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289200);