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 | #2537 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4285 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)(4,6)(5,8)(25,26,31,34)(27,30,32,28)(29,33,39,40)(35,38,37,36);(1,2,4,6)(3,5,8,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,22)(20,21)(23,24)(25,27)(26,30)(28,34)(29,37)(31,32)(33,36)(35,39)(38,40);(1,2)(3,5)(4,6)(7,8)(10,15)(11,17)(18,24)(19,21)(25,28,31,30)(26,32,34,27)(29,38,39,36)(33,35,40,37);(1,3)(2,5)(4,8)(6,7)(9,11,14,17)(10,16,15,12)(13,21,23,19)(18,20,24,22)(25,29,31,39)(26,33,34,40)(27,35,32,37)(28,36,30,38);(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,24,21)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,39)(38,40);(9,13,14,23)(10,18,15,24)(11,19,17,21)(12,20,16,22)(25,30,31,28)(26,32,34,27)(29,36,39,38)(33,35,40,37)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=gi,e^2=ghi,f^2=gi,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, 8633874);