Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×D4×SD32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×D4×D8 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4,5,6);(4,6);(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22);(8,14)(9,21)(10,12)(11,19)(13,17)(16,22)(18,20)110 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×#509, D4×#2187, C22×D4×SD32, … |
| maximal normal subgroup of | C2×D4×#509, D4×#2187, C22×D4×SD32, … |
| maximal quotient of | C2×D4×#357, C2×D4×#367, C2×#499713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530792);