Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2555 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1512 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)(11,15)(12,14)(13,16)(17,18,23,24,22,26,19,21)(20,25,29,31,30,32,27,28);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,19,22,23)(18,24,26,21)(20,27,30,29)(25,31,32,28);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,21,19,26,22,24,23,18)(20,28,27,32,30,31,29,25);(1,5,6,4)(2,7,3,8)(17,19,22,23)(18,24,26,21)(20,29,30,27)(25,28,32,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6331782);