Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
3240 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)(6,8)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(27,33)(28,35)(30,36)(31,38);(10,14)(12,21)(15,17)(18,24)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(5,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,2)(3,4)(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,27,29,33)(26,30,32,36)(28,34,35,39)(31,37,38,40);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,28)(26,31)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=fh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4221665);