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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10,15,19)(11,20,23,16)(12,18,22,17)(13,14,21,24)(25,26,32,36)(27,37,39,30)(28,38,40,33)(29,35,31,34);(1,2,5,8)(3,6,4,7)(9,11,15,23)(10,16,19,20)(12,21,22,13)(14,17,24,18)(25,27,32,39)(26,30,36,37)(28,35,40,34)(29,33,31,38);(1,3,5,4)(2,6,8,7)(9,12,15,22)(10,17,19,18)(11,21,23,13)(14,16,24,20)(25,28,32,40)(26,33,36,38)(27,34,39,35)(29,37,31,30);(9,12)(10,18)(11,21)(13,23)(14,16)(15,22)(17,19)(20,24)(25,29,32,31)(26,34,36,35)(27,33,39,38)(28,37,40,30);(1,2)(3,6)(4,7)(5,8)(9,13,15,21)(10,14,19,24)(11,22,23,12)(16,18,20,17)(25,30)(26,27)(28,29)(31,40)(32,37)(33,34)(35,38)(36,39);(1,4,5,3)(2,7,8,6)(9,14)(10,13)(11,17)(12,16)(15,24)(18,23)(19,21)(20,22)(25,31,32,29)(26,35,36,34)(27,38,39,33)(28,30,40,37)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9117589);