Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×SD162 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×D42 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6,7,8,9,10);(4,6)(5,9)(8,10);(11,12,13,14,15,16,17,18);(12,14)(13,17)(16,18)98 conjugacy classes — view the full character table.
| maximal subgroup of | C2×SD16×AGL(1,ℤ/8), C2×#6362952 |
| maximal normal subgroup of | C2×SD16×AGL(1,ℤ/8), C2×#6362952 |
| maximal quotient of | C2×#244849 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265563);