Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| 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 | |
1740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,7)(10,13)(14,15)(18,21)(22,24)(25,26)(27,28)(29,31)(30,32);(1,2)(3,4)(5,6)(7,8)(9,10,12,13)(11,15,16,14)(18,21)(22,24)(25,26,29,31)(27,28,32,30);(2,6)(4,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,21)(23,24)(25,27,29,32)(26,30,31,28);(1,3,5,8)(2,7,6,4)(9,10)(11,15)(12,13)(14,16)(17,19,20,23)(18,22,21,24)(25,26)(27,30)(28,32)(29,31);(1,4,5,7)(2,3,6,8)(9,11,12,16)(10,14,13,15)(17,18)(19,22)(20,21)(23,24)(25,28,29,30)(26,27,31,32)Polycyclic presentation:
2^9|b^2=fgh,c^2=fghi,d^2=fg,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, 4038118);