Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1540 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,21,24,20,22,28,31)(19,25,27,30,26,32,29,23);(1,3)(2,5)(4,7)(6,8)(9,11,14,10)(12,15,13,16)(18,22)(23,30)(24,31)(25,32);(3,7)(5,8)(9,12,14,13)(10,15,11,16)(17,19,20,26)(18,23,22,30)(21,29,28,27)(24,32,31,25);(3,7)(5,8)(9,13,14,12)(10,15,11,16)(18,22)(23,30)(24,31)(25,32);(2,6)(5,8)(18,22)(23,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfhi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6542834);