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 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,12)(11,13,16,15)(17,18,20,22)(19,23,24,21)(25,26,30,31)(27,28,35,32)(29,37,38,33)(34,40,39,36);(3,8)(5,7)(11,16)(13,15)(17,18,20,22)(19,21,24,23)(26,31)(28,32)(29,38)(34,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,15,12,13)(17,19)(18,21)(20,24)(22,23)(25,27)(26,32)(28,31)(29,39)(30,35)(33,36)(34,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,13,16,15)(25,28,30,32)(26,27,31,35)(29,40,38,36)(33,39,37,34);(1,3)(2,5)(4,8)(6,7)(9,13)(10,11)(12,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,29,30,38)(26,33,31,37)(27,34,35,39)(28,36,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=fh,e^2=f,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, 3965061);