Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | D4×#435 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×D42 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,3,4);(2,4);(5,6)(7,8)(9,10,12,11,13,14,16,15);(6,8);(9,11,12,14,13,15,16,10)140 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#3264, D4×#3306, D4×#3345, … |
| maximal normal subgroup of | D4×#3264, D4×#3306, D4×#3345, … |
| maximal quotient of | #72318337, #72318702, #72318748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291088);