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 | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1632 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)(19,27,31,24)(21,28,30,25)(23,26,29,32);(2,6)(7,8)(17,19,22,31)(18,24,20,27)(21,29,30,23)(25,32,28,26);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,18)(19,28,31,25)(21,27,30,24)(23,26,29,32);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,29)(20,28)(22,30)(23,31)(24,32)(26,27);(1,4,5,3)(2,8,6,7)(19,30)(21,31)(23,29)(24,28)(25,27)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 6496896);