Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 | #36444 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1848 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,18,23,28)(19,29,21,32)(20,27,22,25)(24,31,26,30);(9,12,13,11)(10,14,15,16)(17,19,23,21)(18,24,28,26)(20,30,22,31)(25,29,27,32);(2,6)(5,7)(17,20,23,22)(18,25,28,27)(19,30,21,31)(24,29,26,32);(3,8)(5,7)(17,21,23,19)(18,26,28,24)(20,31,22,30)(25,32,27,29)Polycyclic presentation:
2^9|b^2=fgi,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6597690);