Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#13521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
636 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);(7,8,13,14,20,10,9,15,12,17,21,19,11,16,18,22);(3,4)(5,6)(7,9,11,13,12,18,20,21)(8,14,10,15,17,19,16,22);(4,6)(7,10,11,17,12,16,20,8)(9,15,13,19,18,22,21,14)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406634);