Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,7)(9,10,12,14)(11,13,15,16)(18,21)(22,24)(27,31)(29,32);(1,2,5,6)(3,4,8,7)(11,15)(13,16)(19,23)(22,24)(25,26,28,30)(27,32,31,29);(1,3,5,8)(2,7,6,4)(9,11,12,15)(10,13,14,16)(17,18)(19,24)(20,21)(22,23)(25,26)(27,29)(28,30)(31,32);(1,3)(2,4)(5,8)(6,7)(17,19,20,23)(18,22,21,24)(25,27)(26,29)(28,31)(30,32);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=hi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5959132);