Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1647 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)(3,8)(4,7)(5,6)(11,15)(14,16)(19,23)(21,24)(26,30)(28,35)(31,36)(34,39);(1,3,6,4)(2,7,5,8)(10,13)(14,16)(17,18,20,22)(19,21,23,24)(26,30)(27,33)(32,37)(34,39);(1,3,6,4)(2,8,5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,4,6,3)(2,8,5,7)(17,19)(18,21)(20,23)(22,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,5,6,2)(3,8,4,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4522264);