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 | |
4545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,19)(11,20,22,16)(12,21,24,17)(13,14,18,23)(25,26)(27,32)(28,34)(29,38)(30,36)(31,37)(33,39)(35,40);(1,2,4,6)(3,5,8,7)(9,11,15,22)(10,16,19,20)(12,23,24,14)(13,21,18,17)(25,27,31,30)(26,32,37,36)(28,35,39,38)(29,34,40,33);(1,2)(3,5)(4,6)(7,8)(25,28)(26,33)(27,38)(29,36)(30,35)(31,39)(32,40)(34,37);(1,3,4,8)(2,5,6,7)(9,12,15,24)(10,17,19,21)(11,14,22,23)(13,16,18,20)(25,28)(26,34)(27,35)(29,36)(30,38)(31,39)(32,40)(33,37);(9,13)(10,14)(11,21)(12,16)(15,18)(17,22)(19,23)(20,24)(25,29)(26,35)(27,34)(28,36)(30,33)(31,40)(32,39)(37,38);(9,14)(10,18)(11,12)(13,19)(15,23)(16,17)(20,21)(22,24)(25,30,31,27)(26,36,37,32)(28,35,39,38)(29,34,40,33)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=g,f^2=h,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, 8599023);