Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1289 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)(11,13)(12,14)(15,16)(17,18,23,24,22,27,19,26)(20,29,30,21,31,25,28,32);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,23)(18,24,27,26)(20,30,31,28)(21,29,32,25);(10,14)(11,15)(17,20)(18,25)(19,28)(21,26)(22,31)(23,30)(24,32)(27,29);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(17,21,19,29,22,32,23,25)(18,20,26,28,27,31,24,30);(1,5,6,4)(2,7,3,8)(17,19,22,23)(18,26,27,24)(20,30,31,28)(21,25,32,29)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dfi,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6328854);