Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | #52129 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,27,22,26,29,21)(19,28,20,31,30,24,32,25);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,30)(18,24,26,28)(20,29,32,23)(21,31,27,25);(10,14)(13,16)(17,20)(18,25)(19,29)(21,24)(22,32)(23,30)(26,31)(27,28);(2,5)(6,8)(19,30)(20,32)(24,28)(25,31);(1,3,4,7)(2,6,5,8)(17,21,29,26,22,27,23,18)(19,25,32,24,30,31,20,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,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^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6309376);