Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1600 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)(3,7,8,5)(9,10)(11,16)(12,15)(13,14)(17,18,22,21)(19,24,20,23)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(3,8)(5,7)(9,11,14,12)(10,15,13,16)(17,19,22,20)(18,23,21,24)(26,30)(27,33)(28,36)(38,40);(11,12)(15,16)(17,18,22,21)(19,23,20,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,11)(10,15,13,16)(17,20,22,19)(18,23,21,24)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,10)(11,15,12,16)(17,21,22,18)(19,23,20,24)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=fh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5899438);