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 | #3506 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,6,8,7)(9,10,17,13)(11,21,25,35)(12,26,31,28)(14,15,20,16)(18,24,32,23)(19,30,33,29)(22,34,39,40)(27,37,38,36);(1,3)(2,6)(4,7)(5,8)(9,11,15,23,17,25,16,24)(10,18,14,21,13,32,20,35)(12,27,29,22,31,38,30,39)(19,36,26,34,33,37,28,40);(1,4,5,2)(3,7,8,6)(9,12,16,30,17,31,15,29)(10,19,20,28,13,33,14,26)(11,22,23,38,25,39,24,27)(18,34,21,37,32,40,35,36);(3,8)(6,7)(9,13,15,20,17,10,16,14)(11,21,24,18,25,35,23,32)(12,19,29,26,31,33,30,28)(22,36,27,40,39,37,38,34);(9,14,15,13,17,20,16,10)(11,21,23,32,25,35,24,18)(12,28,29,19,31,26,30,33)(22,36,38,34,39,37,27,40)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,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, 7511698);