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) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3484 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)(6,8)(9,10,13,14)(11,17,19,15)(12,20,22,16)(18,23,21,24)(28,35)(31,38)(33,39)(36,40);(10,14)(15,17)(16,20)(23,24)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,36,39,40);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(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)(12,21,22,18)(16,24,20,23)(25,27,29,34)(26,30,32,37)(28,33,35,39)(31,36,38,40);(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=fgi,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^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4329284);