Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C43 · Schur multiplier (H2) = C210×C42 |
| direct product | C22×C4×#95 |
| subgroup | G/H | |
|---|---|---|
| center | C25×C4 | C22 |
| commutator | C2 | C22×C43 |
| Frattini, cosocle | C24 | C25 |
| socle, prime radical | C26 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
17522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6,7,8);(9,10)(11,12)(13,14,15,16);(10,12)(17,18,19,20)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420516);