Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #331655 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10,12,17)(11,18,19,14)(13,15,21,16)(20,23,24,22)(26,31)(27,35)(33,34)(37,38);(1,3,5,7)(2,6,8,4)(9,11)(10,14)(12,19)(13,20)(15,22)(16,23)(17,18)(21,24)(25,26,29,31)(27,36,35,40)(28,34,30,33)(32,38,39,37);(1,4,5,6)(2,7,8,3)(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(25,27)(26,32)(28,37)(29,35)(30,38)(31,39)(33,36)(34,40);(2,8)(3,7)(10,16)(11,19)(13,21)(14,22)(15,17)(18,23)(25,28,29,30)(26,33,31,34)(27,37,35,38)(32,36,39,40)Polycyclic presentation:
2^9|a^2=ei,b^2=eh,c^2=ehi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 246033);