Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1737 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)(3,5)(4,6)(7,8)(11,15)(14,16)(17,18)(19,20)(21,24)(22,23)(28,35)(32,37)(33,39)(38,40);(9,10,13,12)(11,14,15,16)(18,23)(20,24)(26,30)(31,36)(32,37)(38,40);(1,2,4,6)(3,5,7,8)(9,11,13,15)(10,14,12,16)(17,19,22,21)(18,24,23,20)(25,26,29,30)(27,31,34,36)(28,32,35,37)(33,38,39,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,24)(18,19,23,21)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,16,15,14)(17,21,22,19)(18,24,23,20)(25,28)(26,32)(27,33)(29,35)(30,37)(31,38)(34,39)(36,40)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4837202);