Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,7,8)(3,4,5,6)(11,15)(13,16)(17,18,20,22)(19,21,23,24)(25,26,29,30)(27,33,35,31)(28,32,36,38)(34,40,37,39);(1,3,7,5)(2,4,8,6)(9,10,12,14)(11,13,15,16)(19,23)(21,24)(26,30)(28,36)(31,33)(34,37);(1,4,7,6)(2,3,8,5)(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,21,22,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,5,7,3)(2,4,8,6)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,36)(38,39);(1,6,7,4)(2,3,8,5)(9,11)(10,13)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5690559);