Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | |
1842 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)(11,15,16,12)(17,18,20,22)(19,23,24,21)(25,26)(27,32)(28,30)(29,31);(1,3)(2,5)(4,8)(6,7)(10,14)(11,16)(25,27,29,28)(26,30,31,32);(2,4)(3,7)(9,11,13,16)(10,15,14,12)(17,18)(19,23)(20,22)(21,24)(25,28,29,27)(26,30,31,32);(1,4,6,2)(3,8,7,5)(9,10)(11,15)(12,16)(13,14)(17,19,20,24)(18,21,22,23)(25,26)(27,32)(28,30)(29,31);(1,5,6,8)(2,7,4,3)(9,12,13,15)(10,16,14,11)Polycyclic presentation:
2^9|a^2=gh,b^2=fgi,c^2=f,d^2=g,e^2=h,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, 4837490);