Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16500 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2509 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,23)(22,24)(25,26,31,29)(27,28,32,30);(2,3)(6,7)(9,11,14,16)(10,15,12,13)(17,19,22,21)(18,23,24,20)(25,27)(26,30)(28,29)(31,32);(2,3)(6,7)(10,12)(11,16)(17,19,22,21)(18,23,24,20)(25,28,31,30)(26,32,29,27);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,16,13)(17,20,22,23)(18,19,24,21)(25,29,31,26)(27,28,32,30);(1,4,5,8)(2,6,3,7)(9,13,14,15)(10,16,12,11)(17,21,22,19)(18,23,24,20)(25,30,31,28)(26,32,29,27)Polycyclic presentation:
2^9|a^2=fg,b^2=fhi,c^2=fi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4363497);