Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28 |
| direct product | D4×#340 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4);(2,4);(5,6)(7,8)(9,10,11,12)(14,18)(15,16)(19,20);(6,8)(13,14,16,19,17,20,15,18)140 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#1762, D4×#1879, D4×#1864, … |
| maximal normal subgroup of | D4×#1762, D4×#1879, D4×#1864, … |
| maximal quotient of | D4×#1383, D4×#1386 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263680);