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 | |
4342 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)(11,14)(12,13)(15,16)(17,24)(18,19)(20,22)(21,23)(25,26,31,36)(27,34,39,32)(28,30,35,40)(29,38,37,33);(1,2)(3,8)(4,6)(5,7)(9,11,15,17)(10,14,16,24)(12,20,21,18)(13,22,23,19)(25,27)(26,32)(28,37)(29,35)(30,38)(31,39)(33,40)(34,36);(1,3,5,6)(2,4,7,8)(10,16)(11,17)(12,21)(19,22)(25,28,31,35)(26,30,36,40)(27,37,39,29)(32,38,34,33);(1,2,5,7)(3,4,6,8)(9,12,15,21)(10,13,16,23)(11,18,17,20)(14,19,24,22)(25,29)(26,33)(27,28)(30,32)(31,37)(34,40)(35,39)(36,38);(1,2)(3,4)(5,7)(6,8)(9,13)(10,12)(11,19)(14,18)(15,23)(16,21)(17,22)(20,24)(25,27)(26,34)(28,37)(29,35)(30,33)(31,39)(32,36)(38,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,17)(11,16)(12,19)(13,20)(15,24)(18,23)(21,22)(25,30)(26,35)(27,38)(28,36)(29,32)(31,40)(33,39)(34,37)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10157512);