Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,5,4,7)(17,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,30)(31,32);(1,3,6,4)(2,7,8,5)(9,10,13,14)(11,15,16,12)(17,19,22,28)(18,23,24,30)(20,26,25,31)(21,27,29,32);(9,11)(10,12)(13,16)(14,15)(18,24)(20,25)(23,30)(26,31);(2,8)(3,4)(10,14)(12,15)(17,20,22,25)(18,21,24,29)(19,26,28,31)(23,27,30,32);(1,4,6,3)(2,7,8,5)(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32);(1,5,6,7)(2,4,8,3)(9,11)(10,15)(12,14)(13,16)(17,21)(18,25)(19,27)(20,24)(22,29)(23,31)(26,30)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10133194);