Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | D4×#322 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator | C22 | C22×C4×C8 |
| socle | C23 | C22×C42 |
| prime radical | C22×D4 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1120 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);(5,6,7,10,8,9,11,12);(6,9)(10,12)(13,14,15,17,16,18,19,20)200 conjugacy classes — view the full character table.
| maximal subgroup of | #72216237, #72216350, D4×#1580, … |
| maximal normal subgroup of | #72216237, #72216350, D4×#1580, … |
| maximal quotient of | #72216232, #72216237, D4×#1313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91039);