Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×#107 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(6,8)(9,10,15,22,17,25,35,24)(11,26,29,40,31,37,12,32)(13,23,16,20,14,36,33,21)(18,30,19,28,39,34,38,27);(2,4)(9,11,16,30,15,29,14,28,17,31,33,34,35,12,13,27)(10,18,23,32,24,38,21,37,25,39,36,40,22,19,20,26);(5,6,7,8)(9,12,33,28,15,11,13,34,17,29,16,27,35,31,14,30)(10,19,36,37,24,18,20,40,25,38,23,26,22,39,21,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54287);