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 | |
4401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,20)(11,21,22,16)(12,17,19,14)(13,18,24,23)(25,26,31,36)(27,34,38,32)(28,39,35,30)(29,37,40,33);(1,2)(3,8)(4,6)(5,7)(11,22)(13,24)(16,21)(18,23)(25,27,31,38)(26,32,36,34)(28,29,35,40)(30,37,39,33);(1,3)(2,4)(5,6)(7,8)(9,11,15,22)(10,16,20,21)(12,24,19,13)(14,23,17,18)(25,28,31,35)(26,30,36,39)(27,29,38,40)(32,37,34,33);(3,6)(4,8)(9,12,15,19)(10,17,20,14)(11,13,22,24)(16,23,21,18)(25,29,31,40)(26,33,36,37)(27,35,38,28)(30,34,39,32);(1,4,5,8)(2,6,7,3)(9,13)(10,18)(11,12)(14,16)(15,24)(17,21)(19,22)(20,23)(25,27)(26,34)(28,29)(30,33)(31,38)(32,36)(35,40)(37,39);(1,2)(3,4)(5,7)(6,8)(9,14,15,17)(10,19,20,12)(11,23,22,18)(13,21,24,16)(25,30)(26,35)(27,37)(28,36)(29,34)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068945);