Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1696 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(19,21)(23,24)(25,26,29,30)(27,31,34,36)(28,32,35,37)(33,38,39,40);(9,11,14,13)(10,15,12,16)(17,18,22,20)(19,24,21,23)(26,30)(28,35)(31,36)(33,39);(2,6)(3,8)(10,12)(15,16)(17,19,22,21)(18,23,20,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(1,3,5,8)(2,7,6,4)(9,12,14,10)(11,16,13,15)(17,20,22,18)(19,23,21,24)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,4,5,7)(2,8,6,3)(9,13,14,11)(10,16,12,15)(17,21,22,19)(18,23,20,24)(25,28,29,35)(26,32,30,37)(27,33,34,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=f,b^2=fhi,c^2=fh,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4858597);