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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,11)(12,15,16,14)(17,18,20,22)(19,23,24,21)(25,26,29,31)(27,32,30,28);(1,2,5,6)(3,4,8,7)(10,11)(14,15)(19,24)(21,23)(27,30)(28,32);(2,6)(4,7)(9,10,13,11)(12,15,16,14)(17,19)(18,21)(20,24)(22,23)(25,27,29,30)(26,28,31,32);(1,3,5,8)(2,7,6,4)(9,11,13,10)(12,14,16,15)(17,18)(19,21)(20,22)(23,24);(1,4,5,7)(2,3,6,8)(9,12,13,16)(10,14,11,15)(17,19)(18,21)(20,24)(22,23)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=hi,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, 4038087);