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) = C212 |
| direct product | D4×#386 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×D42 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
11218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(14,16);(13,14)(15,16);(2,4)(5,6,7,8)(10,12);(5,6)(7,8)(9,10,11,12);(1,2)(3,4)(5,6)(7,8)140 conjugacy classes — view the full character table.
| maximal subgroup of | #72232038, #72232039, #72232042, … |
| maximal normal subgroup of | #72232038, #72232039, #72232042, … |
| maximal quotient of | #72251006, D4×#1904, D4×#1910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 426167);