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 | |
2012 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)(11,13)(12,14)(15,16)(18,19)(20,24)(25,26,29,28)(27,31,32,30);(1,3,6,4)(2,7,5,8)(9,10,12,14)(11,13,15,16)(18,19)(22,23);(3,4)(7,8)(11,15)(13,16)(17,18,21,19)(20,22,24,23)(25,26,29,28)(27,30,32,31);(1,4,6,3)(2,7,5,8)(9,11,12,15)(10,13,14,16)(17,19,21,18)(20,23,24,22)(25,27,29,32)(26,30,28,31);(1,5,6,2)(3,7,4,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,24)(18,22,19,23)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=i,d^2=fi,e^2=ghi,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, 1814423);