Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C28 |
| direct product | C2×#11298 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle, prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(8,9)(11,12,14,13)(15,16,17,19,18,20,21,22);(3,4)(5,6)(7,8)(9,10)(11,13,14,12);(4,6)(7,9,10,8)(12,13)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265517);