Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
| direct product | C22×#1381 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C24 | #107 |
| prime radical | C25 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2);(3,4);(5,6)(7,8)(9,10,13,18,15,21,27,20)(11,22,24,36,26,37,38,31)(12,19,14,17,28,34,29,35)(16,25,30,39,33,40,32,23);(6,8)(9,11,14,25,13,24,28,39,15,26,29,40,27,38,12,23)(10,16,19,31,20,32,35,37,21,33,34,36,18,30,17,22)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400055);