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 | |
4522 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)(3,4)(5,6)(7,8)(9,10,15,17)(11,16,21,23)(12,18,19,13)(14,20,24,22)(25,26,32,36)(27,31,35,40)(28,33,39,37)(29,34,38,30);(2,6)(3,7)(9,11)(10,16)(12,22)(13,24)(14,18)(15,21)(17,23)(19,20)(25,27,32,35)(26,31,36,40)(28,30,39,34)(29,37,38,33);(1,3,5,7)(2,4,6,8)(10,17)(11,21)(13,18)(20,22)(25,28,32,39)(26,33,36,37)(27,34,35,30)(29,31,38,40);(1,4)(2,7)(3,6)(5,8)(9,12,15,19)(10,18,17,13)(11,20,21,22)(14,16,24,23)(25,29)(26,34)(27,37)(28,31)(30,36)(32,38)(33,35)(39,40);(9,13,15,18)(10,19,17,12)(11,14,21,24)(16,22,23,20)(25,30,32,34)(26,29,36,38)(27,28,35,39)(31,33,40,37);(9,14)(10,20)(11,18)(12,23)(13,21)(15,24)(16,19)(17,22)(25,31)(26,35)(27,36)(28,38)(29,39)(30,33)(32,40)(34,37)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=h,e^2=gi,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, 8538015);