Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10)(11,14)(12,15)(13,16)(18,19)(20,21)(25,26,30,31)(27,32,34,28)(29,33,37,39)(35,40,38,36);(2,5)(6,8)(10,11)(15,16)(17,18,22,19)(20,23,21,24)(26,31)(27,34)(29,37)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,15,13,16)(17,19,22,18)(20,23,21,24)(25,27)(26,32)(28,31)(29,38)(30,34)(33,36)(35,37)(39,40);(9,12,14,13)(10,15,11,16)(17,20,22,21)(18,23,19,24)(25,28,30,32)(26,27,31,34)(29,36,37,40)(33,35,39,38);(1,3)(2,6)(4,7)(5,8)(9,13,14,12)(10,16,11,15)(17,21,22,20)(18,24,19,23)(25,29,30,37)(26,33,31,39)(27,35,34,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5389339);