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 | |
4423 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)(11,19)(12,21)(13,20)(14,18)(15,16)(17,23)(22,24)(25,26,31,34)(27,29,32,37)(28,30,33,39)(35,36,38,40);(1,3,6,8)(2,7,4,5)(9,11,15,20)(10,13,16,19)(12,22,23,14)(17,24,21,18)(25,27,31,32)(26,29,34,37)(28,35,33,38)(30,36,39,40);(2,4)(3,8)(10,16)(13,19)(17,21)(18,24)(27,32)(29,37)(35,38)(36,40);(3,8)(5,7)(9,12,15,23)(10,17,16,21)(11,14,20,22)(13,18,19,24)(25,28)(26,30)(27,35)(29,36)(31,33)(32,38)(34,39)(37,40);(1,4,6,2)(3,7,8,5)(9,13)(10,11)(12,18)(14,17)(15,19)(16,20)(21,22)(23,24)(25,29,31,37)(26,32,34,27)(28,36,33,40)(30,38,39,35);(1,5,6,7)(2,8,4,3)(9,14)(10,18)(11,12)(13,17)(15,22)(16,24)(19,21)(20,23)(25,30)(26,33)(27,36)(28,34)(29,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,d^2=g,e^2=i,f^2=gh,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, 8529160);