Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#36185 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#555 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(4,6)(7,8,11,13)(9,10,12,14)(16,21)(17,22);(7,9)(8,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,21);(3,4,5,6)(7,10)(8,12)(9,13)(11,14)(15,17,20,22)(16,18,21,19);(7,9)(8,10)(11,12)(13,14)(15,18,20,19)(16,17,21,22);(7,10)(8,12)(9,13)(11,14)(15,19,20,18)(16,22,21,17)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537018);