Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | D4×AΣL(1,𝔽16) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | D4×#120 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C2×D42 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2)(3,4);(2,4);(5,6)(7,11,12,9)(8,10);(6,9)(10,11)65 conjugacy classes — view the full character table.
| maximal subgroup of | #72232721, D4×#2196, #72234132, … |
| maximal normal subgroup of | #72232721, D4×#2196, #72234132, … |
| maximal quotient of | #72232755, C2×D4×AΣL(1,𝔽16), #72317688 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404483);