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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10,13,16)(11,14,12,15)(17,18)(19,21)(20,23)(22,24)(25,26,29,28)(27,30,31,32);(3,7)(6,8)(9,11,13,12)(10,14,16,15)(18,19)(22,23)(27,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,19,21,18)(20,22,24,23)(25,27,29,31)(26,30,28,32);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,15)(13,16);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,20,21,24)(18,22,19,23)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgi,c^2=fghi,d^2=fg,e^2=fghi,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, 2470967);