Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13266 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,15,20,16,19,29,32,17,21,30,35,14,28,23,38)(13,22,25,36,26,34,39,41,27,37,40,42,24,18,31,33);(4,7)(8,9)(11,13,23,31,14,24,30,40,17,27,29,39,16,26,15,25)(12,18,32,36,28,37,20,33,21,34,38,42,19,22,35,41);(3,5,6,10)(4,8,7,9)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402529);