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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,15,16,13)(19,23)(21,24)(25,26,29,30)(27,33,34,31)(28,32,37,36)(35,39,40,38);(11,16)(13,15)(17,18,20,22)(19,21,23,24)(26,30)(27,34)(32,36)(35,40);(9,11)(10,13)(12,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,21,22,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=f,e^2=hi,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, 4688676);