Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28 |
| direct product | D4×#342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,4);(2,4);(9,10)(11,15,16,13)(12,14);(5,6,7,8)(9,11)(10,13)(12,16)(14,15)110 conjugacy classes — view the full character table.
| maximal subgroup of | #72235072, D4×#2021, #72235081, … |
| maximal normal subgroup of | #72235072, D4×#2021, #72235081, … |
| maximal quotient of | #72234290, #72312111, #72312472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263685);