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 | |
1900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,5)(3,8,4,7)(9,10,14,12)(11,16,15,13)(17,18,20,22)(19,23,24,21)(25,26,28,30)(27,31,32,29);(2,5)(3,4)(9,11,14,15)(10,13,12,16)(17,18)(19,23)(20,22)(21,24)(25,26,28,30)(27,31,32,29);(1,3,6,4)(2,7,5,8)(9,10,14,12)(11,16,15,13)(17,18,20,22)(19,21,24,23)(25,26)(27,29)(28,30)(31,32);(1,4,6,3)(2,8,5,7)(9,12,14,10)(11,13,15,16)(17,19,20,24)(18,21,22,23);(1,5,6,2)(3,7,4,8)(9,13)(10,15)(11,12)(14,16)(17,18)(19,23)(20,22)(21,24)(25,27,28,32)(26,29,30,31)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=f,d^2=f,e^2=g,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, 4033459);