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 |
| prime radical | #61085 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1965 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)(11,23)(12,29)(13,19)(14,32)(15,21)(16,20)(17,22)(18,26)(24,35)(25,36)(27,38)(28,37)(30,39)(31,34)(33,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,26,17,28,16,27)(10,18,20,23,22,38,21,37)(12,30,19,24,32,40,31,36)(13,33,29,35,34,39,14,25);(2,5)(6,7)(9,12,16,31,17,32,15,19)(10,13,21,14,22,34,20,29)(11,24,26,40,28,36,27,30)(18,35,23,39,38,25,37,33);(1,3,4,8)(2,6,5,7)(9,13,15,29,17,34,16,14)(10,12,20,19,22,32,21,31)(11,25,27,39,28,35,26,33)(18,36,37,40,38,24,23,30);(9,14,15,13,17,29,16,34)(10,19,20,32,22,31,21,12)(11,25,26,33,28,35,27,39)(18,24,23,40,38,36,37,30)Polycyclic presentation:
2^9|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, 7504261);