Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C82 · Schur multiplier (H2) = C23×C4×C8 |
| direct product | C8×#336 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C8 | C22 |
| commutator | C2 | C4×C82 |
| Frattini, cosocle | C22×C42 | C23 |
| socle, prime radical | C24 | C2×C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,11,13,12,14,15,16);(1,2)(3,4)(17,18,19,21,20,22,23,24);(2,4)(5,6,7,8)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12533);