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 | #41172 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4324 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)(3,7)(4,5)(6,8)(9,10,13,14)(11,17,19,15)(12,16,22,20)(18,24,21,23)(25,26,31,35)(27,32,36,39)(28,33,37,38)(29,30,34,40);(2,5)(6,7)(10,14)(15,17)(16,20)(23,24)(25,27,31,36)(26,32,35,39)(28,29,37,34)(30,38,40,33);(1,2)(3,6)(4,5)(7,8)(10,14)(15,17)(16,20)(23,24)(27,36)(28,37)(32,39)(33,38);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,28,31,37)(26,33,35,38)(27,29,36,34)(30,39,40,32);(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,29)(26,30)(27,37)(28,36)(31,34)(32,38)(33,39)(35,40);(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,30)(26,34)(27,38)(28,32)(29,35)(31,40)(33,36)(37,39)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625626);