Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | |
1565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,8,5,7)(10,13)(11,15)(17,18,21,19)(20,23,24,22)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,39,38,40);(1,2,6,4)(3,8,5,7)(11,15)(14,16)(26,30)(28,37)(31,33)(34,38);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22)(25,27,29,35)(26,31,30,33)(28,38,37,34)(32,40,36,39);(1,4,6,2)(3,8,5,7)(17,20,21,24)(18,22,19,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,40)(38,39);(1,5,6,3)(2,8,4,7)(9,11,12,15)(10,14,13,16)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=hi,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4024869);