Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1785 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)(3,6)(4,7)(5,8)(9,10,13,15)(11,12,16,14)(17,18,21,20)(19,23,24,22);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,15,12)(18,20)(22,23)(26,30)(31,32);(3,8)(5,6)(10,15)(11,16)(17,18,21,20)(19,23,24,22)(25,26,29,30)(27,31,28,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,20,23)(25,27,29,28)(26,31,30,32);(1,2)(3,6)(4,7)(5,8)(9,12,13,14)(10,11,15,16)(17,20,21,18)(19,23,24,22)(25,28,29,27)(26,32,30,31)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fghi,d^2=fghi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1689334);