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×#34192 |
| 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 | |
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)(8,11)(12,14)(16,19)(17,23)(22,26)(24,29);(15,16,20,19)(17,21,23,25)(18,26,27,22)(24,30,29,28);(7,8)(9,12)(10,11)(13,14)(15,17)(16,21)(18,24)(19,25)(20,23)(22,28)(26,30)(27,29);(15,18,20,27)(16,22,19,26)(17,24,23,29)(21,28,25,30);(7,9,10,13)(8,12,11,14)(15,19,20,16)(17,25,23,21)(18,26,27,22)(24,30,29,28)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532623);