Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,14,16)(11,13,12,15)(18,19)(20,24)(27,28)(31,32);(9,11)(10,15)(12,14)(13,16)(18,19)(22,23)(26,30)(31,32);(10,16)(11,12)(17,18,21,19)(20,22,24,23)(25,26)(27,31)(28,32)(29,30);(1,2)(3,5)(4,6)(7,8)(9,12,14,11)(10,13,16,15)(17,19,21,18)(20,23,24,22)(25,27,29,28)(26,31,30,32);(1,3,4,7)(2,5,6,8)(9,13)(10,11)(12,16)(14,15)(17,20,21,24)(18,22,19,23)(25,28,29,27)(26,32,30,31)Polycyclic presentation:
2^9|a^2=fhi,c^2=fg,d^2=h,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4461041);