Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(11,15)(14,16)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(1,3,5,8)(2,6,7,4)(17,18,21,20)(19,23,24,22)(26,30)(27,34)(32,36)(35,40);(2,7)(3,8)(9,10,13,12)(11,14,15,16)(17,19,21,24)(18,22,20,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,39)(38,40);(1,4,5,6)(2,8,7,3)(9,11)(10,14)(12,16)(13,15)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,2)(3,6)(4,8)(5,7)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,23,24,22)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=ghi,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2439939);