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 | |
1920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,22)(21,23)(25,26,29,31)(27,28,32,30);(2,3)(6,7)(10,14)(11,16)(17,19,22,21)(18,23,20,24)(25,27)(26,28)(29,32)(30,31);(2,3)(6,7)(25,27,29,32)(26,30,31,28);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,15,14,12)(17,20,22,18)(19,23,21,24)(25,27)(26,28)(29,32)(30,31);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,16,14,11)(17,21,22,19)(18,23,20,24)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=f,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, 4836638);