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 | |
1681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,6,8,7)(9,10,17,13)(11,21,25,34)(12,26,31,36)(14,15,20,16)(18,24,32,23)(19,30,28,29)(22,33,39,40)(27,37,38,35);(1,3)(2,6)(4,7)(5,8)(9,11,15,23,17,25,16,24)(10,18,14,21,13,32,20,34)(12,27,29,22,31,38,30,39)(19,35,26,33,28,37,36,40);(1,4,5,2)(3,7,8,6)(9,12,16,30,17,31,15,29)(10,19,20,36,13,28,14,26)(11,22,23,38,25,39,24,27)(18,33,21,37,32,40,34,35);(3,8)(6,7)(9,13,15,20,17,10,16,14)(11,21,24,18,25,34,23,32)(12,28,29,36,31,19,30,26)(22,37,27,33,39,35,38,40);(9,14,15,13,17,20,16,10)(11,21,23,32,25,34,24,18)(12,26,29,28,31,36,30,19)(22,37,38,40,39,35,27,33)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=h,d^2=g,e^2=f,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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511686);