Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1544 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18,24,19)(20,25,32,29)(21,26,23,28)(22,27,31,30);(1,3,5,4)(2,6,7,8)(17,19,24,18)(20,27,32,30)(21,26,23,28)(22,25,31,29);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,31)(22,23)(24,32)(26,30)(27,28);(1,4,5,3)(2,6,7,8)(17,21)(18,26)(19,28)(22,31)(23,24)(27,30);(17,22,24,31)(18,27,19,30)(20,23,32,21)(25,28,29,26)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6493491);