Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(19,23)(22,24)(25,26,29,32)(27,30,34,39)(28,35,37,31)(33,40,36,38);(2,5)(6,8)(10,12)(11,15)(18,21)(22,24)(25,27,29,34)(26,30,32,39)(28,36,37,33)(31,40,35,38);(9,10,13,12)(11,14,15,16)(17,18,20,21)(19,22,23,24)(25,28,29,37)(26,31,32,35)(27,33,34,36)(30,38,39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(25,27)(26,30)(28,36)(29,34)(31,40)(32,39)(33,37)(35,38);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32)(33,38)(34,39)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5867710);