Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 | #36062 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1698 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)(12,15)(13,16)(17,18,23,24)(19,25,21,27)(20,30,32,26)(22,28,31,29);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,19)(18,25)(20,22)(21,23)(24,27)(26,29)(28,30)(31,32);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,16,14)(17,20,23,32)(18,26,24,30)(19,29,21,28)(22,25,31,27);(9,12,13,11)(10,15,16,14)(17,21,23,19)(18,27,24,25)(20,31,32,22)(26,28,30,29);(1,4,5,3)(2,6,8,7)(17,22)(18,28)(19,26)(20,25)(21,30)(23,31)(24,29)(27,32)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=fg,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,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, 6498255);