Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
3184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,3,4)(5,6,7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(26,30)(31,37)(32,35)(38,39);(10,13)(15,17)(16,20)(22,23)(25,26,29,30)(27,31,34,37)(28,35,36,32)(33,39,40,38);(6,8)(9,11,14,19)(10,15,13,17)(12,18,21,24)(16,22,20,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(1,2)(3,4)(5,6)(7,8)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,28,29,36)(26,32,30,35)(27,33,34,40)(31,38,37,39)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4849431);