Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C4 · Schur multiplier (H2) = C221 |
| direct product | C24×C4×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C25×C4 | C22 |
| commutator | C2 | C26×C4 |
| Frattini, cosocle | C22 | C27 |
| socle | C26 | C23 |
| prime radical | C25×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
106994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 16 points:
(1,2);(3,4);(5,6);(7,8);(9,10,11,12);(13,14,15,16);(14,16)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493066);