Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C8 · Schur multiplier (H2) = C214 |
| direct product | C23×C8×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C8 | C22 |
| commutator | C2 | C25×C8 |
| Frattini, cosocle | C2×C4 | C26 |
| socle, prime radical | C25 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
9314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(3,4);(5,6);(7,8,9,10,11,12,13,14);(15,16,17,18)(19,22,21,20);(15,19,17,21)(16,20,18,22)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481406);