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 | |
4163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,18)(14,16)(15,17)(19,22)(20,23)(21,24)(25,26,31,35)(27,33,37,29)(28,32,38,40)(30,34,39,36);(3,8)(6,7)(9,11,15,14)(10,16,17,13)(12,20,21,19)(18,22,24,23)(25,27,31,37)(26,29,35,33)(28,36,38,34)(30,32,39,40);(1,3,5,8)(2,6,4,7)(10,17)(12,21)(13,16)(19,20)(25,28,31,38)(26,32,35,40)(27,36,37,34)(29,39,33,30);(1,3)(2,6)(4,7)(5,8)(9,12,15,21)(10,18,17,24)(11,19,14,20)(13,22,16,23)(25,27)(26,33)(28,36)(29,35)(30,32)(31,37)(34,38)(39,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,16)(10,11,17,14)(12,22,21,23)(18,19,24,20)(25,29,31,33)(26,27,35,37)(28,30,38,39)(32,34,40,36);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,13,17,16)(12,20,21,19)(18,22,24,23)(25,30)(26,34)(27,32)(28,29)(31,39)(33,38)(35,36)(37,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gh,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947890);