Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| 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 | #25087 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,25,21,24)(23,26,32,30)(27,29,31,28);(2,6)(7,8)(17,19,22,21)(18,24,20,25)(23,28,32,29)(26,31,30,27);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,18)(19,27,21,31)(23,26,32,30)(24,28,25,29);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,25,20,24)(23,29,32,28)(26,27,30,31);(1,4,5,3)(2,8,6,7)(19,28)(21,29)(23,32)(24,31)(25,27)(26,30)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=fg,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6496842);