Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1858 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)(9,10,13,12)(11,15,16,14)(17,18,22,21)(19,20,24,23)(25,26,31,28)(27,29,32,30);(1,3,6,5)(2,7,4,8)(10,12)(14,15)(19,24)(20,23)(25,27)(26,30)(28,29)(31,32);(1,2,6,4)(3,7,5,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,24)(18,23,21,20)(25,28,31,26)(27,29,32,30);(1,4,6,2)(3,8,5,7)(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,24)(22,23)(25,29,31,30)(26,27,28,32);(1,5,6,3)(2,8,4,7)(9,11)(10,14)(12,15)(13,16)(17,21,22,18)(19,20,24,23)(25,30,31,29)(26,32,28,27)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=fg,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 758362);