Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,7)(5,6)(9,10,13,16)(11,14,12,15)(17,18)(19,24)(20,23)(21,22)(25,26,28,29)(27,30,31,32);(1,3,6,4)(2,7,5,8)(9,11,13,12)(10,14,16,15)(17,19)(18,20)(21,24)(22,23)(26,29)(30,32);(2,5)(7,8)(18,21)(19,23)(25,27,28,31)(26,30,29,32);(1,4,6,3)(2,8,5,7)(9,10)(11,14)(12,15)(13,16)(17,20,22,24)(18,23,21,19)(25,26)(27,30)(28,29)(31,32);(1,5,6,2)(3,8,4,7)(9,12,13,11)(10,15,16,14)(17,21,22,18)(19,24,23,20)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgi,c^2=fg,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2423791);