Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(3,8)(10,14)(11,15)(17,18,22,20)(19,24,23,21)(27,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,15)(13,14)(17,18,22,20)(19,24,23,21);(1,3,5,8)(2,6,4,7)(9,11,13,15)(10,12,14,16)(17,19)(18,21)(20,24)(22,23)(25,26,28,30)(27,29,31,32);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,11,14,15)(17,20,22,18)(19,24,23,21)(25,27,28,31)(26,29,30,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,12)(13,15)(14,16)(17,21,22,24)(18,23,20,19)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fghi,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4018469);