Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C210×C4 |
| direct product | C22×#1735 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C25 | C2×D4 |
| prime radical | C26 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
9058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2);(3,4);(5,6)(7,8)(9,10,11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,28)(26,27);(6,8)(13,15,17,25,18,26,16,24)(14,19,20,27,22,28,21,23);(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,27,28,23)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276943);