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 | |
1682 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)(10,17)(11,22)(12,28)(13,30)(14,15)(18,32)(19,35)(20,21)(24,38)(26,27)(33,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,10,14,17,16,21,15,20)(11,23,26,32,27,34,22,18)(12,29,13,19,30,31,28,35)(24,33,25,37,38,40,36,39);(1,4,5,2)(3,7,8,6)(9,11,14,26,16,27,15,22)(10,18,20,34,21,32,17,23)(12,24,13,25,30,38,28,36)(19,33,29,39,35,40,31,37);(3,8)(6,7)(9,12,15,28,16,30,14,13)(10,19,17,31,21,35,20,29)(11,24,22,36,27,38,26,25)(18,33,23,37,32,40,34,39);(9,13,15,12,16,28,14,30)(10,19,20,29,21,35,17,31)(11,25,22,24,27,36,26,38)(18,33,34,39,32,40,23,37)Polycyclic presentation:
2^9|a^2=gh,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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511687);