Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10,14,12)(11,13,16,15)(19,23)(22,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,15,12,13)(18,21)(22,24)(26,31)(27,32)(29,37)(35,40);(2,5)(3,7)(10,12)(13,15)(17,18)(19,22)(20,21)(23,24)(25,27,30,32)(26,28,31,36)(29,38,37,34)(33,40,39,35);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,13,16,15)(25,28)(26,32)(27,31)(29,35)(30,36)(33,38)(34,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,11)(12,16)(14,15)(17,19,20,23)(18,22,21,24)(25,29,30,37)(26,33,31,39)(27,34,32,38)(28,35,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=fg,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5426841);