Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#34326 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7566 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);(3,4,6,5)(7,8,11,10)(9,13,14,12)(15,16)(17,24)(18,21)(19,27)(20,23)(22,28)(25,29)(26,30);(15,17,20,18)(16,21,23,24)(19,25,28,26)(22,29,27,30);(7,9,11,14)(8,12,10,13)(15,18,20,17)(16,21,23,24)(19,26,28,25)(22,29,27,30);(7,8,11,10)(9,12,14,13)(15,16)(17,24)(18,21)(19,22)(20,23)(25,30)(26,29)(27,28);(3,5,6,4)(7,10,11,8)(9,13,14,12)(15,19)(16,22)(17,25)(18,26)(20,28)(21,29)(23,27)(24,30)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532784);