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 | |
4151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,5)(3,4,8,6)(9,10)(11,12)(13,18)(14,21)(15,17)(16,20)(19,23)(22,24)(25,26,29,32)(27,30,34,38)(28,35,36,31)(33,39,40,37);(2,5)(3,8)(9,11,15,20)(10,16,17,12)(13,23,24,14)(18,21,22,19)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,3)(2,6)(4,5)(7,8)(10,17)(12,16)(18,22)(19,21)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4,7,6)(2,3,5,8)(9,12)(10,11)(13,19)(14,22)(15,16)(17,20)(18,23)(21,24);(1,5,7,2)(3,4,8,6)(9,13,15,24)(10,18,17,22)(11,14,20,23)(12,21,16,19)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,6,7,4)(2,8,5,3)(9,14)(10,19)(11,13)(12,22)(15,23)(16,18)(17,21)(20,24)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,d^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9117558);