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 | |
4608 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)(9,10)(11,14)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26,32,35)(27,36,39,33)(28,40,30,31)(29,37,38,34);(1,3,6,5)(2,7,4,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27,32,39)(26,33,35,36)(28,37,30,34)(29,31,38,40);(10,11)(15,17)(16,18)(22,23)(25,28,32,30)(26,31,35,40)(27,37,39,34)(29,33,38,36);(2,4)(7,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,29,32,38)(26,34,35,37)(27,31,39,40)(28,36,30,33);(1,4,6,2)(3,8,5,7)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,30,32,28)(26,31,35,40)(27,37,39,34)(29,36,38,33);(1,5,6,3)(2,7,4,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,31)(26,30)(27,38)(28,35)(29,39)(32,40)(33,34)(36,37)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=hi,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10102649);