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 | |
4417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10,14,13)(11,17,19,15)(12,20,22,16)(18,23,21,24)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(2,5)(3,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,24)(21,22)(28,36)(31,38)(34,40)(37,39);(1,3,6,8)(2,7,5,4)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4)(2,3)(5,8)(6,7)(9,12,14,22)(10,16,13,20)(11,18,19,21)(15,23,17,24);(1,5,6,2)(3,4,8,7)(9,13,14,10)(11,17,19,15)(12,16,22,20)(18,23,21,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,17)(12,21)(13,15)(14,19)(16,23)(18,22)(20,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9131538);