Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1596 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)(3,7)(4,6)(5,8)(10,14)(12,15)(19,23)(22,24)(25,26,30,31)(27,32,34,28)(29,33,38,39)(35,40,37,36);(1,3,6,5)(2,7,4,8)(10,14)(11,16)(18,21)(22,24)(26,31)(27,34)(33,39)(35,37);(1,2,6,4)(3,8,5,7)(9,10)(11,15)(12,16)(13,14)(17,18,20,21)(19,22,23,24)(25,27)(26,32)(28,31)(29,37)(30,34)(33,36)(35,38)(39,40);(1,4,6,2)(3,7,5,8)(9,11,13,16)(10,15,14,12)(25,28,30,32)(26,27,31,34)(29,36,38,40)(33,35,39,37);(1,5,6,3)(2,7,4,8)(9,12,13,15)(10,16,14,11)(17,19)(18,22)(20,23)(21,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fghi,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4557694);