Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2185 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)(11,16)(12,17)(13,21)(14,15)(18,23)(19,20)(22,24)(27,33)(28,35)(30,36)(31,38);(10,15)(11,17)(18,21)(19,24)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(5,6,7,8)(9,11,14,17)(10,16,15,12)(13,19,23,24)(18,22,21,20)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,2)(3,4)(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,21,24)(25,27,29,33)(26,30,32,36)(28,34,35,39)(31,37,38,40);(9,13,14,23)(10,18,15,21)(11,19,17,24)(12,20,16,22)(25,28)(26,31)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40)Polycyclic presentation:
2^9|b^2=fh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4326655);