Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2184 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)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(26,31)(27,34)(33,37)(35,40);(6,8)(11,18)(14,17)(19,24)(22,23)(25,26,30,31)(27,28,34,32)(29,37,39,33)(35,38,40,36);(5,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,27,30,34)(26,32,31,28)(29,35,39,40)(33,38,37,36);(1,2)(3,4)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,40,37,35);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,29,30,39)(26,33,31,37)(27,35,34,40)(28,36,32,38)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2978716);