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 | |
1877 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,22)(21,23)(25,26)(27,29)(28,30)(31,32);(2,6)(3,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,21)(18,23,20,24);(2,6)(3,8)(10,13)(14,15)(18,20)(23,24)(25,26,28,30)(27,29,31,32);(1,3,5,8)(2,7,6,4)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,23,21,24)(25,26)(27,29)(28,30)(31,32);(1,4,5,7)(2,8,6,3)(9,11,12,16)(10,14,13,15)(17,21,22,19)(18,23,20,24)(25,27,28,31)(26,29,30,32)Polycyclic presentation:
2^9|b^2=i,c^2=fgh,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, 4836816);