Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#126 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26)(27,33)(28,38)(29,34)(30,40)(31,36)(32,37)(35,39);(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27,31,28)(26,33,36,38)(29,30,32,39)(34,40,37,35);(2,5)(6,8)(25,28,32,39,31,27,29,30)(26,33,37,40,36,38,34,35);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29,31,32)(26,34,36,37)(27,30,28,39)(33,40,38,35);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,30,32,28,31,39,29,27)(26,35,37,33,36,40,34,38)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=h,d^2=fg,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6735785);