Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1439 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)(17,18,22,27)(19,24,29,32)(20,30,28,23)(21,25,31,26);(1,3,5,4)(2,6,7,8)(10,14)(12,15)(17,19,22,29)(18,23,27,30)(20,31,28,21)(24,26,32,25);(9,10)(11,12)(13,14)(15,16)(17,20,22,28)(18,24,27,32)(19,21,29,31)(23,25,30,26);(1,4,5,3)(2,6,7,8)(9,11,13,16)(10,15,14,12)(17,18)(19,23)(20,32)(21,26)(22,27)(24,28)(25,31)(29,30);(9,12,13,15)(10,11,14,16)(17,21,22,31)(18,25,27,26)(19,20,29,28)(23,24,30,32)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=i,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6531676);