Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533087 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(11,17)(14,18)(19,23)(21,24)(25,26,28,32,29,33,37,30)(27,34,35,39,36,40,38,31);(1,3,5,4)(2,6,8,7)(9,10,13,16)(11,18,17,14)(12,15,20,22)(19,24,23,21)(26,30)(28,37)(31,34)(32,33)(35,38)(39,40);(1,3,5,4)(2,7,8,6)(9,11,13,17)(10,14,16,18)(12,19,20,23)(15,21,22,24)(25,27,29,36)(26,31,33,39)(28,38,37,35)(30,34,32,40);(1,4,5,3)(2,6,8,7)(12,20)(15,22)(19,23)(21,24);(9,12,13,20)(10,15,16,22)(11,19,17,23)(14,21,18,24)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=fg,d^2=fghi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6470544);