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 | |
4172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23)(25,26)(27,32)(28,36)(29,34)(30,38)(31,35)(33,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,17,19,14)(12,15,22,20)(18,24,21,23)(25,27,31,30)(26,32,35,38)(28,37,39,34)(29,36,40,33);(1,3)(2,7)(4,6)(5,8)(9,10,13,16)(11,17,19,14)(12,15,22,20)(18,24,21,23)(25,28,31,39)(26,33,35,36)(27,34,30,37)(29,32,40,38);(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,28)(26,33)(27,34)(29,38)(30,37)(31,39)(32,40)(35,36);(1,4,5,2)(3,7,8,6)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,29)(26,34)(27,33)(28,38)(30,36)(31,40)(32,39)(35,37);(1,3)(2,7)(4,6)(5,8)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(25,30,31,27)(26,32,35,38)(28,37,39,34)(29,33,40,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947927);