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 | |
2038 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)(10,17)(11,22)(12,28)(13,31)(14,15)(18,34)(19,20)(21,32)(24,30)(26,27)(35,40)(36,37);(1,2)(3,6)(4,5)(7,8)(9,10,14,17,16,21,15,32)(11,23,26,34,27,38,22,18)(12,29,13,20,31,33,28,19)(24,36,39,40,30,35,25,37);(2,5)(6,7)(9,11,15,22,16,27,14,26)(10,18,17,23,21,34,32,38)(12,30,28,39,31,24,13,25)(19,36,29,40,20,35,33,37);(1,3,4,8)(2,6,5,7)(9,12,14,13,16,31,15,28)(10,19,32,33,21,20,17,29)(11,24,26,39,27,30,22,25)(18,35,38,40,34,36,23,37);(9,13,15,12,16,28,14,31)(10,20,32,29,21,19,17,33)(11,25,22,30,27,39,26,24)(18,35,38,40,34,36,23,37)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=gi,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=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504235);