Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2462 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)(3,7)(4,5)(6,8)(10,13)(11,15)(18,21)(22,24)(25,26,29,32)(27,31,28,30);(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16)(19,23)(22,24)(25,27,29,28)(26,30,32,31);(2,5)(6,7)(11,15)(14,16)(17,18,20,21)(19,22,23,24)(25,26)(27,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,18)(19,22)(20,21)(23,24)(25,28,29,27)(26,31,32,30);(1,3)(2,6)(4,8)(5,7)(9,10)(11,16)(12,13)(14,15)(17,19)(18,22)(20,23)(21,24)(25,26)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=fghi,b^2=fghi,c^2=fi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5690898);