Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1884 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)(3,7,8,4)(11,15)(14,16)(18,21)(19,23)(25,26,29,32)(27,31,28,30);(2,6)(3,8)(10,13)(14,16)(25,27,29,28)(26,30,32,31);(1,3,5,8)(2,7,6,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,21)(22,23)(25,28,29,27)(26,31,32,30);(1,2)(3,7)(4,8)(5,6)(17,19)(18,22)(20,23)(21,24)(25,26)(27,31)(28,30)(29,32);(1,4,5,7)(2,8,6,3)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=f,e^2=h,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, 4837245);