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 | #43504 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4308 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)(3,8)(4,5)(6,7)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,20,23,24)(25,26,31,34)(27,30,32,28)(29,33,39,40)(35,38,37,36);(1,3,6,5)(2,4,7,8)(9,11,13,18)(10,14,16,21)(12,17,19,23)(15,20,22,24)(25,27)(26,30)(28,34)(29,37)(31,32)(33,36)(35,39)(38,40);(1,3,6,5)(2,4,7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,28,31,30)(26,32,34,27)(29,38,39,36)(33,35,40,37);(2,7)(3,5)(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,23)(22,24)(25,29,31,39)(26,33,34,40)(27,35,32,37)(28,36,30,38);(1,4,6,8)(2,3,7,5)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,39)(38,40);(1,5,6,3)(2,4,7,8)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)(25,30,31,28)(26,32,34,27)(29,36,39,38)(33,35,40,37)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gi,d^2=i,e^2=g,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, 8627335);