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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1535 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,29,22)(20,25,30,32)(21,26,28,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,23,29)(18,24,27,22)(20,25,30,32)(21,31,28,26);(10,14)(13,16)(17,20,23,30)(18,25,27,32)(19,28,29,21)(22,26,24,31);(1,3,4,7)(2,6,5,8)(19,29)(20,26)(21,32)(22,24)(25,28)(30,31);(17,21,23,28)(18,26,27,31)(19,30,29,20)(22,25,24,32)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=i,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, 6487401);