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) = 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 | |
2220 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)(9,10,14,15)(11,16,17,12)(13,18,22,23)(19,24,21,20)(25,26,30,33)(27,28,35,31)(29,37,39,32)(34,40,38,36);(10,15)(12,16)(18,23)(20,24)(25,27,30,35)(26,31,33,28)(29,34,39,38)(32,40,37,36);(5,6)(7,8)(9,11,14,17)(10,16,15,12)(13,21,22,19)(18,20,23,24)(25,27)(26,31)(28,33)(29,38)(30,35)(32,36)(34,39)(37,40);(1,2,3,4)(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,28,30,31)(26,27,33,35)(29,36,39,40)(32,34,37,38);(9,13,14,22)(10,18,15,23)(11,19,17,21)(12,20,16,24)(25,29,30,39)(26,32,33,37)(27,34,35,38)(28,36,31,40)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=i,e^2=f,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, 4296658);