Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,8,4)(10,12)(11,13)(18,23)(19,21)(27,31)(30,32);(1,3,5,8)(2,6,7,4)(9,10)(11,16)(12,14)(13,15)(17,18,22,23)(19,20,21,24)(26,29)(30,32);(2,7)(3,8)(9,11,14,13)(10,15,12,16)(17,19,22,21)(18,24,23,20)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,8)(5,7)(9,12,14,10)(11,15,13,16)(17,20,22,24)(18,19,23,21);(1,4,5,6)(2,8,7,3)(9,13,14,11)(10,15,12,16)(17,21,22,19)(18,24,23,20)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4837239);