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 | |
4550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,18)(12,21)(13,16)(14,15)(17,22)(19,23)(20,24)(25,26,31,29)(27,32,37,38)(28,34,39,33)(30,36,40,35);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(25,27,31,37)(26,32,29,38)(28,35,39,36)(30,33,40,34);(9,12,15,22)(10,17,14,21)(11,19,13,20)(16,23,18,24)(25,28,31,39)(26,33,29,34)(27,36,37,35)(30,38,40,32);(1,3,5,8)(2,6,4,7)(9,12)(10,17)(11,20)(13,19)(14,21)(15,22)(16,24)(18,23)(25,28)(26,34)(27,36)(29,33)(30,38)(31,39)(32,40)(35,37);(1,3)(2,7)(4,6)(5,8)(9,13,15,11)(10,16,14,18)(12,20,22,19)(17,23,21,24)(25,29,31,26)(27,32,37,38)(28,34,39,33)(30,35,40,36);(1,4,5,2)(3,7,8,6)(9,14,15,10)(11,18,13,16)(12,21,22,17)(19,24,20,23)(25,30)(26,35)(27,33)(28,38)(29,36)(31,40)(32,39)(34,37)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=ghi,e^2=h,f^2=g,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, 10154464);