Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C22×C8 |
| direct product | C8×Q8⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | D4 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C43 | C23 |
| socle, prime radical | C23 | C4×C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8);(10,13)(11,15)(14,16)(17,18,19,21,20,22,23,24);(9,10,12,14)(11,16,15,13)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3881);