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 | |
4556 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)(4,6)(5,8)(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,21)(23,24)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,37,39);(1,3,6,8)(2,7,4,5)(10,15)(11,19)(12,22)(21,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,38)(39,40);(2,4)(3,8)(10,15)(12,22)(16,18)(20,24)(29,38)(32,40)(34,37)(36,39);(3,8)(5,7)(9,11,14,19)(10,16,15,18)(12,20,22,24)(13,23,17,21)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(1,4,6,2)(3,7,8,5)(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,23)(19,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37);(1,5,6,7)(2,8,4,3)(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,24,18,20)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,d^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, 8529136);