Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(10,14)(12,15)(17,18,23,22)(19,20,24,21)(25,26,29,31)(27,32,30,28);(2,4)(3,8)(9,10)(11,12)(13,14)(15,16)(17,19,23,24)(18,21,22,20)(27,30)(28,32);(2,4)(6,7)(10,14)(12,15)(17,20,23,21)(18,19,22,24)(25,27,29,30)(26,28,31,32);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,15,14,12)(17,21,23,20)(18,24,22,19);(1,4,5,2)(3,6,8,7)(9,12,13,15)(10,11,14,16)(17,22,23,18)(19,21,24,20)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fhi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038089);