Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
2359 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)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,16,12)(17,18)(19,20)(21,23)(22,24)(27,31)(30,32);(1,2,4,6)(3,5,7,8)(9,11)(10,12)(13,16)(14,15)(17,19,22,23)(18,21,24,20);(3,7)(5,8)(10,14)(12,15)(17,20,22,21)(18,19,24,23)(25,26,29,28)(27,30,31,32);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,14)(13,16)(17,21,22,20)(18,23,24,19)(25,27)(26,30)(28,32)(29,31);(1,3,4,7)(2,5,6,8)(9,12,13,15)(10,16,14,11)(17,18)(19,21)(20,23)(22,24)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=hi,d^2=h,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3597403);