Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#10829 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #381 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(7,8)(9,15,17,12)(10,18)(11,14)(13,19)(16,20,22,21);(3,4,5,6)(7,9)(8,12)(10,16)(11,17)(13,20)(14,15)(18,21)(19,22);(7,10)(8,13,14,18)(9,16)(11,19)(12,20,15,21)(17,22)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260748);