Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(11,15)(13,16)(17,18,20,22)(19,21,23,24)(25,26,30,31)(27,28,33,32)(29,37,39,34)(35,38,40,36);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(26,31)(27,33)(34,37)(35,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,33)(26,32,31,28)(29,35,39,40)(34,38,37,36);(9,11)(10,13)(12,15)(14,16)(17,19,20,23)(18,21,22,24)(25,28,30,32)(26,33,31,27)(29,38,39,36)(34,35,37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(25,29,30,39)(26,34,31,37)(27,35,33,40)(28,36,32,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4330571);