Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,8,5,7)(10,12)(11,15)(17,18,20,22)(19,21,23,24)(26,30)(28,36)(31,34)(35,39);(2,4)(3,5)(9,10,14,12)(11,16,15,13)(19,23)(21,24)(26,30)(27,33)(32,37)(35,39);(1,3,6,5)(2,7,4,8)(9,11)(10,13)(12,16)(14,15)(17,19,20,23)(18,21,22,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,4,6,2)(3,8,5,7)(9,12,14,10)(11,16,15,13)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,5,6,3)(2,8,4,7)(9,13,14,16)(10,15,12,11)(17,18)(19,21)(20,22)(23,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=i,c^2=fghi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5959102);