Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,8,4,7)(10,13)(14,15)(17,18)(19,21)(20,22)(23,24)(26,30)(27,33)(28,36)(38,40);(1,3,6,4)(2,7,5,8)(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,23,24,21)(26,30)(28,36)(31,34)(35,39);(2,5)(3,4)(9,11,12,16)(10,14,13,15)(17,19,20,24)(18,21,22,23)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,4,6,3)(2,8,5,7)(9,10)(11,14)(12,13)(15,16)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,5,6,2)(3,8,4,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,22)(21,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,c^2=fg,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393462);