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 | #2416 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,14,17,24)(12,13,21,22)(18,23,20,19)(25,26,31,32)(27,35,30,33)(28,38,34,29)(36,40,39,37);(1,2,6,4)(3,8,5,7)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(26,32)(28,34)(33,35)(36,39);(1,3,6,5)(2,7,4,8)(10,16)(11,17)(12,21)(19,23)(25,27,31,30)(26,33,32,35)(28,39,34,36)(29,37,38,40);(3,5)(7,8)(9,12)(10,13)(11,18)(14,23)(15,21)(16,22)(17,20)(19,24)(25,28,31,34)(26,29,32,38)(27,36,30,39)(33,40,35,37);(1,4,6,2)(3,8,5,7)(9,13)(10,12)(11,19)(14,20)(15,22)(16,21)(17,23)(18,24)(25,29)(26,34)(27,37)(28,32)(30,40)(31,38)(33,36)(35,39);(1,5,6,3)(2,7,4,8)(9,14,15,24)(10,17,16,11)(12,19,21,23)(13,18,22,20)(25,30,31,27)(26,33,32,35)(28,36,34,39)(29,37,38,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,d^2=g,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, 8633781);