Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(10,13)(11,15)(19,23)(21,24)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(10,13)(14,16)(17,18,20,22)(19,21,23,24)(26,31)(27,34)(29,37)(36,39);(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,19,20,23)(18,21,22,24)(25,28,30,32)(26,27,31,34)(29,36,37,39)(33,35,38,40);(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=g,d^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 605221);