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 | |
1618 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)(3,7,8,5)(9,10,13,14)(11,15,16,12)(19,23)(21,24)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(25,27,30,35)(26,31,33,28)(29,34,38,37)(32,39,40,36);(3,8)(5,7)(11,16)(12,15)(17,18,20,22)(19,21,23,24)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19,20,23)(18,21,22,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,15)(14,16)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fi,d^2=fgi,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, 4604969);