Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| 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 | |
1930 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)(3,5,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,20)(19,24,21,23)(25,26)(27,30)(28,29)(31,32);(1,3)(2,7)(4,5)(6,8)(9,10,13,16)(11,15,12,14)(19,21)(23,24)(26,29)(27,31);(3,8)(5,7)(9,11,13,12)(10,14,16,15)(17,19,22,21)(18,23,20,24)(25,26,28,29)(27,30,31,32);(1,4,6,2)(3,7,8,5)(9,12,13,11)(10,15,16,14)(17,20,22,18)(19,24,21,23)(25,27,28,31)(26,30,29,32);(1,5,6,7)(2,3,4,8)(9,10)(11,15)(12,14)(13,16)(17,21,22,19)(18,24,20,23)(25,26)(27,32)(28,29)(30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=g,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4035647);