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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2144 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)(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,20)(19,24,21,23)(25,26,29,30)(27,32,31,28);(9,11,13,12)(10,14,15,16)(18,20)(23,24)(26,30)(27,31);(2,6)(3,8)(10,15)(14,16)(17,19,22,21)(18,23,20,24)(25,26,29,30)(27,28,31,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,20,22,18)(19,23,21,24)(25,27,29,31)(26,28,30,32);(1,4,5,7)(2,3,6,8)(9,12,13,11)(10,16,15,14)(17,21,22,19)(18,23,20,24)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=f,d^2=f,e^2=g,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, 4836537);