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) = 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 | |
1608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(10,14)(12,16)(17,18,20,21)(19,23,24,22);(3,8)(5,7)(10,14)(11,15)(18,21)(22,23)(25,26,29,28)(27,31,32,30);(1,3)(2,5)(4,8)(6,7)(9,10,13,14)(11,12,15,16)(17,18)(19,23)(20,21)(22,24)(25,27,29,32)(26,30,28,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,12,14,16)(25,26,29,28)(27,30,32,31);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,19,20,24)(18,22,21,23)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=fhi,d^2=fhi,e^2=fgh,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, 2340870);