Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13104 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C22 | C23×C16 |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C24 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16,20,25,21,17,26,27,19,24,28,30,18,23,22,29);(8,11)(12,14)(15,17,20,27,21,24,26,30,19,23,28,29,18,16,22,25);(4,6)(7,9,10,13)(8,12,11,14)(15,18,19,21)(16,23,24,17)(20,22,28,26)(25,29,30,27)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401958);