Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1524 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)(3,5)(4,7)(6,8)(9,10,13,14)(11,12,16,15)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40);(1,2,5,3)(4,7,8,6)(10,14)(11,16)(18,21)(22,24)(26,30)(31,38)(32,35)(39,40);(2,3)(4,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(1,3,5,2)(4,6,8,7)(9,12,13,15)(10,16,14,11)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,4)(2,6)(3,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,19,20,23)(18,22,21,24)(25,28,29,37)(26,32,30,35)(27,33,34,36)(31,39,38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4107169);