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 |
| prime radical | #51424 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(12,20)(15,21)(18,24)(22,23)(25,26,32,35)(27,30,34,40)(28,37,39,33)(29,31,38,36);(1,3,6,5)(2,7,4,8)(9,10)(11,16)(12,21)(13,17)(14,19)(15,20)(18,22)(23,24)(25,27,32,34)(26,30,35,40)(28,36,39,31)(29,33,38,37);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,28,32,39)(26,33,35,37)(27,36,34,31)(29,30,38,40);(1,2,6,4)(3,8,5,7)(9,12,13,20)(10,15,17,21)(11,18,19,24)(14,22,16,23)(25,29)(26,31)(27,33)(28,30)(32,38)(34,37)(35,36)(39,40);(1,4,6,2)(3,8,5,7)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,30)(26,34)(27,35)(28,38)(29,39)(31,33)(32,40)(36,37);(1,5,6,3)(2,7,4,8)(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,31)(26,29)(27,28)(30,33)(32,36)(34,39)(35,38)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10388262);