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 | |
1582 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)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,21,19)(20,23,24,22)(25,26,30,31)(27,32,34,28)(29,33,39,40)(35,38,37,36);(2,4)(6,7)(10,13)(11,15)(17,19,21,18)(20,23,24,22)(26,31)(27,34)(33,40)(35,37);(2,4)(3,8)(9,11,14,15)(10,12,13,16)(25,27)(26,32)(28,31)(29,37)(30,34)(33,36)(35,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,16)(10,15,13,11)(17,20,21,24)(18,22,19,23)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,37,40,35);(1,4,5,2)(3,6,8,7)(9,13,14,10)(11,12,15,16)(25,29,30,39)(26,33,31,40)(27,35,34,37)(28,36,32,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=fgh,d^2=fi,e^2=f,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, 4151911);