Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #52307 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1632 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)(9,10,12,14)(11,15,16,13)(17,18,22,25,21,20,24,29)(19,26,28,32,27,23,30,31);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,13,14,15);(2,6)(3,7)(11,16)(13,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32);(2,6)(5,8)(10,14)(11,16)(19,27)(23,26)(28,30)(31,32);(17,20,22,29,21,18,24,25)(19,23,28,31,27,26,30,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=cfgi,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315178);