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 | |
4512 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)(11,17)(12,13)(14,22)(15,16)(18,23)(19,21)(20,24)(25,26)(27,30)(28,37)(29,32)(31,34)(33,36)(35,39)(38,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,12)(10,13,16,17)(14,19,23,20)(18,21,22,24)(25,27)(26,30)(28,38)(29,35)(31,36)(32,39)(33,34)(37,40);(2,6)(3,7)(10,16)(13,17)(18,22)(21,24)(25,28,31,29)(26,32,34,37)(27,35,36,38)(30,40,33,39);(3,7)(4,8)(9,12,15,11)(10,17,16,13)(14,19,23,20)(18,21,22,24)(25,29,31,28)(26,32,34,37)(27,35,36,38)(30,39,33,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,17)(10,12,16,11)(14,21,23,24)(18,20,22,19)(25,30,31,33)(26,27,34,36)(28,39,29,40)(32,38,37,35);(1,4)(2,3)(5,8)(6,7)(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,22)(17,24)(25,27)(26,33)(28,35)(29,38)(30,34)(31,36)(32,39)(37,40)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8972140);