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 |
| prime radical | #59117 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(10,15)(11,19)(17,23)(18,20)(25,26,31,32)(27,34,28,33)(29,30,39,40)(35,38,37,36);(1,2,5,8)(3,4,7,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(26,32)(29,39)(33,34)(35,37);(1,2)(3,6)(4,7)(5,8)(25,27,31,28)(26,33,32,34)(29,35,39,37)(30,36,40,38);(3,7)(4,6)(9,11,14,19)(10,16,15,21)(12,18,22,20)(13,23,24,17)(25,28,31,27)(26,34,32,33)(29,35,39,37)(30,36,40,38);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,17,15,23)(11,20,19,18)(13,16,24,21)(25,29,31,39)(26,30,32,40)(27,35,28,37)(33,36,34,38);(1,4,5,6)(2,7,8,3)(9,13,14,24)(10,18,15,20)(11,17,19,23)(12,21,22,16)(25,30)(26,29)(27,36)(28,38)(31,40)(32,39)(33,35)(34,37)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9129409);