Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C43 |
| direct product | C22×#2867 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
33858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2);(3,4);(6,8)(10,11)(14,15);(9,10,12,11);(13,14,16,15);(5,6,7,8)(9,11,12,10)(13,15,16,14)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540544);