Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | |
2160 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,16,12)(17,18,22,21)(19,20,24,23)(25,26,30,31)(27,28,29,32);(1,3,6,8)(2,7,4,5)(9,11,13,16)(10,12,15,14)(17,19)(18,23)(20,21)(22,24)(26,31)(28,32);(2,4)(3,8)(9,11,13,16)(10,12,15,14)(17,20,22,23)(18,19,21,24)(25,27,30,29)(26,28,31,32);(1,4,6,2)(3,5,8,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,24)(22,23)(25,28,30,32)(26,27,31,29);(1,5)(2,3)(4,8)(6,7)(9,12,13,14)(10,11,15,16)(17,21,22,18)(19,20,24,23)(25,29,30,27)(26,32,31,28)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836449);