Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16291 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,20,23,22)(27,28)(31,32);(1,3,5,8)(2,6,7,4)(9,11,12,16)(10,13,14,15)(17,19,21,23)(18,22,24,20)(26,30)(31,32);(2,7)(3,8)(10,14)(13,15)(17,20,21,22)(18,23,24,19)(25,26,29,30)(27,31,28,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,20)(18,19)(21,22)(23,24)(25,27,29,28)(26,31,30,32);(1,4,5,6)(2,3,7,8)(9,11)(10,15)(12,16)(13,14)(17,19)(18,20)(21,23)(22,24)(25,28,29,27)(26,32,30,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4431079);