Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
3276 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)(11,17)(12,14)(13,20)(15,18)(16,22)(19,23)(21,24)(25,26,29,32)(27,30,34,38)(28,35,36,31)(33,39,40,37);(10,12)(11,18)(16,20)(19,21)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(5,6)(7,8)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,2)(3,4)(5,6)(7,8)(9,12,14,10)(11,17,18,15)(13,20,22,16)(19,24,21,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,d^2=f,e^2=h,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, 4289193);