Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1692 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)(19,23)(21,24)(25,26,29,30)(27,31,34,36)(28,32,35,37)(33,38,39,40);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,21,23,24)(26,30)(28,35)(31,36)(33,39);(2,6)(3,8)(9,10,12,14)(11,13,16,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,23)(22,24)(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=gh,c^2=fgi,d^2=f,e^2=f,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, 4870303);