Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | #2699 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1404 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)(17,18,23,24,22,27,19,26)(20,30,25,29,31,21,28,32);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,19,22,23)(18,24,27,26)(20,25,31,28)(21,29,30,32);(9,11,12,16)(10,13,14,15)(17,20,22,31)(18,21,27,30)(19,28,23,25)(24,32,26,29);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,21,19,29,22,30,23,32)(18,25,26,20,27,28,24,31);(1,5,6,4)(2,7,3,8)(17,19,22,23)(18,26,27,24)(20,25,31,28)(21,32,30,29)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=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, 6331785);