Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#34059 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| socle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,23)(18,21)(19,22)(20,25)(24,29)(26,28)(27,30);(15,17,19,25)(16,20,22,23)(18,26,27,24)(21,29,30,28);(8,11)(12,14)(15,18,19,27)(16,21,22,30)(17,24,25,26)(20,28,23,29);(16,22)(20,23)(21,30)(28,29);(7,9,10,13)(8,12,11,14)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532469);