Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563327 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,16,21)(11,22,27,29)(12,28,20,15)(13,19,33,30)(14,34,32,18)(17,26,36,25)(23,35,24,39)(31,38,40,37);(1,3)(2,5)(4,7)(6,8)(9,11,15,25,16,27,28,26)(10,17,20,22,21,36,12,29)(13,31,32,23,33,40,14,24)(18,37,19,35,34,38,30,39);(9,12,15,10,16,20,28,21)(11,17,26,29,27,36,25,22)(13,19,32,34,33,30,14,18)(23,39,31,38,24,35,40,37);(3,7)(5,8)(9,13,15,32,16,33,28,14)(10,18,20,19,21,34,12,30)(11,23,26,31,27,24,25,40)(17,35,29,37,36,39,22,38);(9,14,28,33,16,32,15,13)(10,19,12,18,21,30,20,34)(11,24,26,40,27,23,25,31)(17,35,29,37,36,39,22,38)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=g,d^2=g,e^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7500214);