Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1730 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)(3,5)(4,6)(7,8)(9,10,13,14)(11,12,15,16)(17,18)(19,20)(21,23)(22,24)(25,26,29,30)(27,31,28,32);(2,3)(6,7)(9,11)(10,12)(13,15)(14,16)(17,19,22,21)(18,23,24,20)(26,30)(31,32);(2,3)(6,7)(10,14)(11,15)(18,24)(20,23)(25,27,29,28)(26,31,30,32);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,15)(14,16)(17,20,22,23)(18,19,24,21)(25,26)(27,31)(28,32)(29,30);(1,4,5,8)(2,6,3,7)(9,12)(10,11)(13,16)(14,15)(17,21,22,19)(18,23,24,20)(25,28,29,27)(26,32,30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fgh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4837179);