Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(17,18,23,20)(19,28,31,32)(21,26,22,27)(24,29,25,30);(1,3,5,4)(2,6,8,7)(10,14)(12,15)(17,19,21,30,23,31,22,29)(18,24,27,28,20,25,26,32);(1,3,5,4)(2,7,8,6)(9,10)(11,12)(13,14)(15,16)(17,19,22,29,23,31,21,30)(18,25,26,28,20,24,27,32);(1,4,5,3)(2,6,8,7)(9,11,13,16)(10,15,14,12)(17,20,23,18)(19,24,31,25)(21,26,22,27)(28,29,32,30);(9,12,13,15)(10,11,14,16)(17,21,23,22)(18,26,20,27)(19,29,31,30)(24,28,25,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=fi,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6531468);