Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4308740 |
| 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 | |
1573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(4,8)(10,13)(11,15)(18,22)(23,24)(25,26,29,32)(27,28,31,30);(1,2,6,5)(3,4,7,8)(10,13)(14,16)(25,27)(26,30)(28,32)(29,31);(1,3)(2,4)(5,8)(6,7)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,28)(29,31)(30,32);(1,4,6,8)(2,7,5,3)(17,19,21,20)(18,23,22,24)(25,28)(26,31)(27,32)(29,30);(1,5,6,2)(3,8,7,4)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4217103);