Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,12,15,16)(17,18)(19,21)(20,22)(23,24)(25,26,29,30)(27,33,35,31)(28,32,37,36)(34,39,40,38);(9,11,13,15)(10,12,14,16)(17,19,20,23)(18,21,22,24)(26,30)(31,33)(32,36)(38,39);(10,14)(11,15)(17,18,20,22)(19,21,23,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,3,4,7)(2,5,6,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,21)(20,23)(22,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=gh,d^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5500845);