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 | #3036 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1637 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)(10,17)(11,22)(12,28)(13,31)(14,15)(18,35)(19,20)(21,33)(24,30)(26,27)(36,39)(37,38);(1,3)(2,6)(4,7)(5,8)(9,10,14,17,16,21,15,33)(11,23,26,35,27,40,22,18)(12,29,13,20,31,34,28,19)(24,37,25,39,30,36,32,38);(1,4,5,2)(3,7,8,6)(9,11,15,22,16,27,14,26)(10,18,17,23,21,35,33,40)(12,30,28,25,31,24,13,32)(19,37,29,39,20,36,34,38);(3,8)(6,7)(9,12,15,28,16,31,14,13)(10,19,17,29,21,20,33,34)(11,24,22,32,27,30,26,25)(18,36,23,38,35,37,40,39);(9,13,15,12,16,28,14,31)(10,20,33,29,21,19,17,34)(11,25,22,24,27,32,26,30)(18,37,40,38,35,36,23,39)Polycyclic presentation:
2^9|a^2=ghi,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=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, 7511700);