Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,21)(22,24)(26,30)(27,33)(28,36)(38,40);(2,5)(3,8)(9,11,12,16)(10,13,14,15)(17,18,20,21)(19,23,24,22)(26,30)(28,36)(31,34)(35,39);(2,5)(6,7)(10,14)(13,15)(18,21)(22,23)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,16)(13,14)(17,19)(18,22)(20,24)(21,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5135918);