Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13445 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,19)(20,22);(8,14)(9,16)(10,11)(12,19)(15,18)(17,22)(20,21);(4,6)(7,9,10,18,19,22,21,17,13,15,20,16,12,14,11,8)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406376);