Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | |
1862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,8,7,4)(9,10,13,12)(11,15,16,14)(17,18,21,23)(19,22,24,20)(25,26)(27,31)(28,32)(29,30);(1,2,5,6)(3,4,7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,20)(21,23)(22,24)(26,30)(31,32);(2,6)(3,7)(9,11,13,16)(10,14,12,15)(17,19,21,24)(18,20,23,22)(25,27,29,28)(26,31,30,32);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,30);(1,4)(2,7)(3,6)(5,8)(9,12,13,10)(11,15,16,14)(17,20,21,22)(18,19,23,24)(25,28,29,27)(26,32,30,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=h,e^2=f,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, 4836880);