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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1576 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,25)(21,28)(22,27)(24,26)(29,31)(30,32);(1,3,5,4)(2,6,7,8)(17,19,23,18)(20,29,22,32)(21,28,24,26)(25,30,27,31);(9,11,12,16)(10,13,14,15)(17,20,23,22)(18,25,19,27)(21,30,24,31)(26,29,28,32);(1,4,5,3)(2,6,7,8)(17,21)(18,26)(19,28)(20,22)(23,24)(29,32);(17,22,23,20)(18,27,19,25)(21,31,24,30)(26,32,28,29)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=gi,d^2=fi,e^2=i,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, 6493254);