Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #759288 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,8,5,7)(10,14)(11,16)(18,21)(19,23)(25,26,28,30)(27,31,32,29);(1,3,6,5)(2,7,4,8)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,30)(29,32);(1,2,6,4)(3,8,5,7)(9,10)(11,15)(12,16)(13,14)(17,19,20,23)(18,22,21,24)(25,27,28,32)(26,29,30,31);(1,4,6,2)(3,8,5,7)(9,11,13,16)(10,15,14,12);(1,5,6,3)(2,8,4,7)(9,12,13,15)(10,16,14,11)(17,18)(19,24)(20,21)(22,23)(25,26)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,e^c=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 668017);