Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#22464 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#577 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,12,16,13,9,11,19)(10,15,20,17,21,18,22,14);(7,9,13,8)(10,17,21,14)(11,16,12,19)(15,20,18,22);(4,6)(7,10)(8,14)(9,17)(11,20)(12,22)(13,21)(15,19)(16,18);(7,10)(8,15)(9,18)(11,22)(12,20)(13,21)(14,19)(16,17)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286395);