Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #2714 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1463 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)(17,18,24,29,23,28,21,27)(19,22,20,26,30,25,31,32);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(17,19,23,30)(18,25,28,22)(20,21,31,24)(26,29,32,27);(10,14)(13,16)(17,20)(18,26)(19,21)(22,27)(23,31)(24,30)(25,29)(28,32);(1,4,6,5)(2,7,3,8)(17,21,23,24)(18,27,28,29)(19,20,30,31)(22,26,25,32);(1,5,6,4)(2,7,3,8)(17,22,21,32,23,25,24,26)(18,20,27,19,28,31,29,30)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=df,e^b=ei,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6310095);