Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249463 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,14,16,13,18,12,17)(11,19,22,15,21,23,20,24)(25,26,29,30)(27,31,34,38)(28,35,37,32)(33,36,40,39);(2,5)(6,7)(9,11,13,21)(10,15,18,24)(12,22,14,20)(16,19,17,23)(25,27,29,34)(26,31,30,38)(28,36,37,39)(32,33,35,40);(10,16)(12,14)(15,19)(17,18)(20,22)(23,24)(25,28,29,37)(26,32,30,35)(27,33,34,40)(31,39,38,36);(1,3,4,8)(2,6,5,7)(10,17)(12,14)(15,23)(16,18)(19,24)(20,22)Polycyclic presentation:
2^9|a^2=ehi,b^2=hi,c^2=i,d^2=efhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi,e^d=eh,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 219744);