Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#453 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | D4×#453 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 12 points:
(1,2,3,4);(2,4);(6,9)(7,11)(10,12);(5,6)(7,12)(8,10)(9,11);(7,11)(9,12)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215857, #72219652, #72226215, … |
| maximal normal subgroup of | #72215857, #72219652, #72226215, … |
| maximal quotient of | #72226283, #72234127, #72257992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480855);