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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1980 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)(3,7)(4,5)(6,8)(9,10,16,21)(11,22,26,28)(12,27,20,15)(13,30,32,19)(14,18,33,29)(17,25,36,24)(23,34,40,35)(31,38,39,37);(1,2)(3,6)(4,5)(7,8)(9,11,15,24,16,26,27,25)(10,17,20,22,21,36,12,28)(13,31,14,23,32,39,33,40)(18,37,30,34,29,38,19,35);(2,5)(6,7)(9,12,15,10,16,20,27,21)(11,17,25,28,26,36,24,22)(13,30,14,29,32,19,33,18)(23,35,31,38,40,34,39,37);(1,3,4,8)(2,6,5,7)(9,13,15,14,16,32,27,33)(10,18,20,30,21,29,12,19)(11,23,25,31,26,40,24,39)(17,34,28,37,36,35,22,38);(9,14,27,13,16,33,15,32)(10,19,12,29,21,30,20,18)(11,23,25,31,26,40,24,39)(17,35,28,38,36,34,22,37)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,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, 7504234);