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 | |
4341 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)(3,7)(4,6)(5,8)(10,14)(11,17)(16,19)(18,20)(25,26)(27,30)(28,34)(29,39)(31,33)(32,35)(36,37)(38,40);(1,2,4,6)(3,5,8,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,27,32,31)(26,33,35,30)(28,36,29,38)(34,40,39,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,28,32,29)(26,34,35,39)(27,36,31,38)(30,37,33,40);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,29,32,28)(26,34,35,39)(27,36,31,38)(30,40,33,37);(9,11,13,17)(10,15,14,22)(12,20,21,18)(16,24,19,23)(25,30)(26,27)(28,37)(29,40)(31,35)(32,33)(34,36)(38,39);(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,31,32,27)(26,33,35,30)(28,38,29,36)(34,40,39,37)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8607644);