Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,7)(3,4)(10,13)(14,16)(17,18,20,22)(19,23,24,21)(26,30)(27,33)(28,36)(39,40);(1,2)(3,8)(4,5)(6,7)(11,15)(14,16)(19,24)(21,23)(25,26,29,30)(27,34,33,31)(28,32,36,38)(35,40,37,39);(1,3,6,4)(2,8,7,5)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,4,6,3)(2,8,7,5)(9,11)(10,14)(12,15)(13,16)(17,19,20,24)(18,21,22,23)(25,27,29,33)(26,31,30,34)(28,37,36,35)(32,40,38,39);(1,5,6,8)(2,4,7,3)(25,28,29,36)(26,32,30,38)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=fh,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4802671);