Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,15,19)(11,16,14,29)(12,22,24,39)(13,25,28,38)(17,32,34,23)(18,21,36,27)(20,37,30,35)(26,31,40,33);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,16,19,29)(12,20,24,30)(13,21,28,27)(17,31,34,33)(18,25,36,38)(22,37,39,35)(23,26,32,40);(9,12,15,24)(10,17,19,34)(11,20,14,30)(13,26,28,40)(16,31,29,33)(18,35,36,37)(21,32,27,23)(22,38,39,25);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^b=df,g^c=gh,h^c=hi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72234336, #72234341, #72234342, … |
| maximal normal subgroup of | #72234336, #72234341, #72234342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531185);