Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2050 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)(10,11)(12,16)(18,21)(19,23)(25,26,29,32)(27,28,31,30);(1,3,6,4)(2,7,5,8)(10,11)(14,15)(17,18,20,21)(19,24,23,22)(25,27,29,31)(26,30,32,28);(3,4)(7,8)(9,10,13,11)(12,14,16,15)(17,19,20,23)(18,22,21,24)(25,28,29,30)(26,31,32,27);(1,4,6,3)(2,7,5,8)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,23)(21,24)(25,28)(26,27)(29,30)(31,32);(1,5,6,2)(3,7,4,8)(9,12,13,16)(10,14,11,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=i,d^2=fi,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1814418);