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 | |
4334 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)(3,8)(4,6)(5,7)(25,26,32,36)(27,31,35,40)(28,30,39,33)(29,34,38,37);(2,4)(7,8)(9,10,14,15)(11,16,20,22)(12,19,21,17)(13,18,23,24)(25,27,32,35)(26,31,36,40)(28,34,39,37)(29,30,38,33);(1,3,6,5)(2,7,4,8)(10,15)(12,21)(16,22)(18,24)(25,28,32,39)(26,33,36,30)(27,37,35,34)(29,31,38,40);(1,2,6,4)(3,7,5,8)(9,11)(10,16)(12,18)(13,17)(14,20)(15,22)(19,23)(21,24)(25,29)(26,34)(27,33)(28,31)(30,35)(32,38)(36,37)(39,40);(1,4,6,2)(3,7,5,8)(9,12,14,21)(10,17,15,19)(11,18,20,24)(13,22,23,16)(25,30,32,33)(26,28,36,39)(27,29,35,38)(31,37,40,34);(1,5,6,3)(2,7,4,8)(9,13)(10,18)(11,19)(12,22)(14,23)(15,24)(16,21)(17,20)(25,31)(26,35)(27,36)(28,38)(29,39)(30,37)(32,40)(33,34)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=g,e^2=gi,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, 8525262);