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 | |
4557 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)(3,7)(4,6)(5,8)(9,10,15,18)(11,13,16,21)(12,14,17,23)(19,20,22,24)(26,30)(27,33)(32,38)(35,36);(1,3,6,8)(2,7,4,5)(9,11,15,16)(10,13,18,21)(12,19,17,22)(14,20,23,24)(25,26,29,30)(27,34,33,31)(28,32,37,38)(35,40,36,39);(2,4)(3,8)(11,16)(13,21)(19,22)(20,24)(25,26,29,30)(27,31,33,34)(28,32,37,38)(35,39,36,40);(3,8)(5,7)(9,12)(10,14)(11,19)(13,20)(15,17)(16,22)(18,23)(21,24)(25,27)(26,31)(28,36)(29,33)(30,34)(32,40)(35,37)(38,39);(1,4,6,2)(3,7,8,5)(9,13,15,21)(10,16,18,11)(12,20,17,24)(14,22,23,19)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,40)(37,38);(1,5,6,7)(2,8,4,3)(9,14)(10,17)(11,20)(12,18)(13,22)(15,23)(16,24)(19,21)(25,28,29,37)(26,32,30,38)(27,35,33,36)(31,39,34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=g,e^2=i,f^2=h,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, 8529134);