Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563333 |
| 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 | |
2009 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)(3,7)(4,5)(6,8)(9,10,15,19)(11,20,25,22)(12,26,30,32)(13,31,18,14)(16,24,34,23)(17,35,28,29)(21,33,39,40)(27,37,38,36);(1,2)(3,6)(4,5)(7,8)(9,11,14,23,15,25,31,24)(10,16,18,20,19,34,13,22)(12,27,29,21,30,38,35,39)(17,36,26,33,28,37,32,40);(2,5)(6,7)(9,12,14,29,15,30,31,35)(10,17,18,26,19,28,13,32)(11,21,24,27,25,39,23,38)(16,33,22,36,34,40,20,37);(1,3,4,8)(2,6,5,7)(9,10,14,18,15,19,31,13)(11,20,24,16,25,22,23,34)(12,28,29,32,30,17,35,26)(21,36,27,40,39,37,38,33);(9,13,14,10,15,18,31,19)(11,22,23,16,25,20,24,34)(12,26,29,28,30,32,35,17)(21,37,38,40,39,36,27,33)Polycyclic presentation:
2^9|a^2=i,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,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504262);