Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
3346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,14)(13,20)(16,22)(17,18)(19,24)(21,23)(25,26)(27,34)(28,36)(29,32)(30,33)(31,39)(35,38)(37,40);(10,12)(11,17)(16,20)(19,21)(25,27,30,32)(26,29,33,34)(28,37,39,35)(31,40,36,38);(5,6)(7,8)(9,11,14,17)(10,15,12,18)(13,21,22,19)(16,24,20,23)(25,26)(27,29)(28,31)(30,33)(32,34)(35,38)(36,39)(37,40);(1,2)(3,4)(5,6)(7,8)(9,12,14,10)(11,18,17,15)(13,20,22,16)(19,24,21,23)(25,28,30,39)(26,31,33,36)(27,35,32,37)(29,38,34,40);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29)(26,32)(27,33)(28,38)(30,34)(31,37)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=h,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4287690);