Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #52164 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,23,25,22,26,21,20)(19,27,29,28,30,24,31,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,24)(20,28)(21,31)(22,30)(23,29)(25,32)(26,27);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,20,21,26,22,25,23,18)(19,28,31,27,30,32,29,24);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,21,22,23)(18,20,26,25)(19,29,30,31)(24,32,27,28);(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,25,26,20)(19,29,30,31)(24,28,27,32)Polycyclic presentation:
2^9|a^2=h,c^2=fh,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,e^b=ei,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6319556);