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 | |
4610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,5,7)(9,10)(11,16)(12,13)(14,19)(15,17)(18,24)(20,21)(22,23)(25,26)(27,37)(28,34)(29,31)(30,35)(32,36)(33,40)(38,39);(1,3,6,5)(2,7,8,4)(9,11)(10,16)(12,22)(13,23)(14,17)(15,19)(18,20)(21,24)(25,27,32,40)(26,33,36,37)(28,38,29,35)(30,31,39,34);(2,8)(4,7)(10,17)(11,19)(13,24)(20,22)(25,28,32,29)(26,34,36,31)(27,38,40,35)(30,37,39,33);(2,8)(3,5)(9,12,15,18)(10,13,17,24)(11,20,19,22)(14,23,16,21)(25,29,32,28)(26,34,36,31)(27,35,40,38)(30,37,39,33);(1,4,6,7)(2,3,8,5)(9,13)(10,18)(11,21)(12,17)(14,20)(15,24)(16,22)(19,23)(25,30,32,39)(26,35,36,38)(27,34,40,31)(28,33,29,37);(1,5,6,3)(2,7,8,4)(9,14)(10,19)(11,17)(12,21)(13,20)(15,16)(18,23)(22,24)(25,31,32,34)(26,29,36,28)(27,39,40,30)(33,35,37,38)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8510013);