Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#12556 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C24 | C2×D8 |
| prime radical | C25 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1762 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)(5,6)(7,8,9,10)(11,12)(13,24)(14,19)(15,21)(16,20)(17,23)(18,26)(22,30)(25,38)(27,32)(28,39)(29,36)(31,37)(33,41)(34,42)(35,40);(4,6)(11,13,16,27,14,25,29,40,17,28,31,42,30,41,15,26)(12,18,21,33,22,34,37,39,23,35,36,38,19,32,20,24);(11,14,17,30)(12,19,23,22)(13,25,28,41)(15,16,29,31)(18,32,35,34)(20,36,37,21)(24,38,39,33)(26,27,40,42)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400051);