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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#126 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1463 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,24,28,23)(21,25,29,30)(26,31,27,32);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(18,22)(19,26)(21,29)(23,31)(24,32)(27,28);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19,20,28)(18,23,22,24)(21,27,29,26)(25,31,30,32);(1,4,5,3)(2,8,6,7)(19,27)(21,29)(23,31)(24,32)(25,30)(26,28);(17,19,20,28)(18,24,22,23)(21,27,29,26)(25,32,30,31)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,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, 6488007);