Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(17,18,21,19)(20,23,24,22)(25,26)(27,30)(28,32)(29,37)(31,34)(33,39)(35,40)(36,38);(2,5)(3,7)(17,18)(19,21)(20,23)(22,24)(25,27,31,28)(26,32,34,30)(29,35,39,36)(33,40,37,38);(2,5)(6,8)(18,19)(22,23)(25,28,31,27)(26,32,34,30)(29,35,39,36)(33,38,37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,23,24,22)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,13,14,16)(17,20)(18,22)(19,23)(21,24)(25,30,31,32)(26,27,34,28)(29,38,39,40)(33,35,37,36)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=fg,d^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4461123);