Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#34262 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8006 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)(8,11)(12,14)(15,16,21,19)(17,24,18,22)(20,27,28,23)(25,29,26,30);(15,17,21,18)(16,22,19,24)(20,25,28,26)(23,29,27,30);(3,5,6,4)(7,8)(9,12)(10,11)(13,14)(15,18,21,17)(16,22,19,24)(20,26,28,25)(23,29,27,30);(15,19,21,16)(17,22,18,24)(20,27,28,23)(25,29,26,30);(7,9,10,13)(8,12,11,14)(15,20)(16,23)(17,25)(18,26)(19,27)(21,28)(22,29)(24,30)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532652);