Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×AGL(1,ℤ/16) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
777 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(6,10,12,19)(7,14)(8,15)(11,17,20,13)(16,18);(5,6)(7,11)(8,17)(9,12)(10,18)(13,15)(14,19)(16,20);(5,7,15,18,9,16,8,14)(6,11,13,10,12,20,17,19)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400445);