Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #32902 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(17,18,23,20)(19,26,29,32)(21,25,22,30)(24,31,27,28);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19,23,29)(18,24,20,27)(21,28,22,31)(25,32,30,26);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(17,20,23,18)(19,27,29,24)(21,25,22,30)(26,28,32,31);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(18,20)(24,27)(25,30)(26,32);(1,4,5,3)(2,7,8,6)(10,13)(11,15)(17,21,23,22)(18,25,20,30)(19,28,29,31)(24,32,27,26)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gi,d^2=g,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524127);