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 | |
1610 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)(11,15)(12,13)(14,16)(18,21)(22,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,10)(11,15)(12,13)(14,16)(19,23)(22,24)(25,26,29,32)(27,30,34,38)(28,35,36,31)(33,39,40,37);(10,13)(14,15)(17,18,20,21)(19,22,23,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,16)(13,15)(17,19,20,23)(18,22,21,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,35)(34,40)(38,39)Polycyclic presentation:
2^9|a^2=fghi,b^2=f,c^2=fgh,d^2=f,e^2=i,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, 5485458);