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 | |
1619 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)(3,4)(5,8)(6,7)(9,10)(11,15)(12,14)(13,16)(19,23)(22,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,3,6,5)(2,4,7,8)(9,11,14,13)(10,15,12,16)(18,21)(22,24)(26,27)(29,37)(31,33)(35,40);(2,7)(4,8)(9,11,14,13)(10,16,12,15)(17,18)(19,22)(20,21)(23,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,4,6,8)(2,3,7,5)(9,12,14,10)(11,16,13,15)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,5,6,3)(2,8,7,4)(9,13,14,11)(10,16,12,15)(17,19)(18,22)(20,23)(21,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|b^2=fg,c^2=fi,d^2=g,e^2=fg,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, 4869736);