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 | |
1940 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)(5,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18)(19,21)(20,22)(23,24)(25,26,29,30)(27,31,32,28);(1,3)(2,7)(4,8)(5,6)(11,16)(13,15)(17,18,20,22)(19,23,24,21)(26,30)(27,32);(2,8)(3,5)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,21,24,23)(25,26,29,30)(27,31,32,28);(1,4)(2,3)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,20,24)(18,21,22,23)(25,27)(26,31)(28,30)(29,32);(1,5,6,3)(2,7,8,4)(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,22)(21,24)(25,28,29,31)(26,32,30,27)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=ghi,d^2=fh,e^2=h,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, 4038407);