Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(18,21)(22,24)(25,26)(27,30)(28,29)(31,32);(4,8)(6,7)(10,14)(11,16)(17,18,20,21)(19,22,23,24)(25,27,29,32)(26,30,28,31);(2,3)(6,7)(11,16)(12,15)(18,21)(19,23)(26,28)(30,31);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24)(25,28,29,26)(27,31,32,30);(1,4)(2,6)(3,7)(5,8)(9,12,13,15)(10,11,14,16)(17,19,20,23)(18,22,21,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5143679);