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 | |
4333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,14)(12,16)(13,17)(15,18)(19,21)(20,23)(22,24)(25,26)(27,37)(28,40)(29,30)(31,34)(32,36)(33,39)(35,38);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,32,39)(26,33,36,37)(28,38,31,30)(29,34,35,40);(1,3,5,4)(2,6,8,7)(11,18)(12,20)(14,15)(16,23)(25,28,32,31)(26,34,36,40)(27,38,39,30)(29,33,35,37);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,29,32,35)(26,30,36,38)(27,34,39,40)(28,37,31,33);(1,3,5,4)(2,6,8,7)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,30)(26,35)(27,28)(29,36)(31,39)(32,38)(33,40)(34,37);(1,4,5,3)(2,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,31,32,28)(26,34,36,40)(27,30,39,38)(29,33,35,37)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7647755);