Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,8,4,7)(10,13)(11,15)(17,18,20,22)(19,21,23,24)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,2,6,5)(3,7,4,8)(10,13)(14,16)(26,31)(29,38)(32,39)(36,40);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,4,6,3)(2,8,5,7)(25,28,30,37)(26,32,31,39)(27,34,33,35)(29,36,38,40);(1,5,6,2)(3,8,4,7)(9,11,12,15)(10,14,13,16)(17,19,20,23)(18,21,22,24)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fi,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4265423);