Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#16702 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(4,6)(7,8,11,12)(9,13,14,10)(16,17)(20,21)(23,24)(25,29)(26,27)(28,30);(8,12)(9,14)(15,16,19,17)(18,20,22,21)(23,25)(24,28)(26,29)(27,30);(3,4,5,6)(7,9)(8,13)(10,12)(11,14)(15,17,19,16)(18,20,22,21)(23,25)(24,29)(26,28)(27,30);(7,10,11,13)(8,14,12,9)(15,18,19,22)(16,20,17,21)(23,26,27,24)(25,29,30,28)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518628);