Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | |
1790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,21)(19,20,24,23)(27,33)(28,35)(31,36)(32,37);(9,11,14,13)(10,15,12,16)(17,19)(18,23)(20,21)(22,24)(26,30)(31,36)(32,37)(38,40);(10,12)(15,16)(17,20,22,23)(18,19,21,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,16,13,15)(17,20)(18,19)(21,24)(22,23)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,16,12,15)(17,21,22,18)(19,20,24,23)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=fgh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4839578);