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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,13,16)(11,14,12,15)(18,24)(19,28)(26,30)(27,32);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,18,22,24)(19,29,28,31)(20,30,25,26)(21,27,23,32);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,16,14)(17,19,22,28)(18,25,24,20)(21,30,23,26)(27,31,32,29);(9,12,13,11)(10,15,16,14)(17,20,22,25)(18,26,24,30)(19,27,28,32)(21,31,23,29);(1,4,5,3)(2,6,8,7)(17,21)(19,30)(22,23)(26,28)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fgi,c^2=fgi,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,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, 6498504);