Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C82 · Schur multiplier (H2) = C24×C8 |
| direct product | C82×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C82 | C22 |
| commutator | C2 | C22×C82 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C23 | C22×C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4,5,6,7,8);(9,10,11,12,13,14,15,16);(17,18,19,20)(21,24,23,22);(17,21,19,23)(18,22,20,24)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91567);