Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C4×#2971 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C24×C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4);(5,6)(7,10)(8,9)(11,12)(13,14,18,15,17,19,16,20);(6,9)(10,12)(14,19)(15,20);(13,15,18,19,17,20,16,14);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,15,19,20)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277337);