Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15263 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,15)(11,12,16,14)(18,19)(20,24)(26,28)(30,32);(1,2,4,6)(3,5,7,8)(9,11,13,16)(10,14,15,12)(18,19)(22,23)(27,31)(30,32);(3,7)(5,8)(10,15)(11,16)(17,18,21,19)(20,22,24,23)(25,26,29,28)(27,30,31,32);(1,2)(3,5)(4,6)(7,8)(9,12,13,14)(10,16,15,11)(17,19,21,18)(20,23,24,22)(25,27)(26,30)(28,32)(29,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,20,21,24)(18,22,19,23)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=ghi,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2456219);