Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2231 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(18,21)(19,20)(25,26,31,32)(27,29,30,28);(1,3,6,5)(2,7,4,8)(9,10,13,12)(11,14,16,15)(17,18,22,21)(19,24,20,23)(25,27,31,30)(26,28,32,29);(3,5)(7,8)(10,12)(14,15)(17,19,22,20)(18,23,21,24)(25,28,31,29)(26,27,32,30);(1,4,6,2)(3,7,5,8)(9,11,13,16)(10,14,12,15)(17,20,22,19)(18,23,21,24)(25,29,31,28)(26,27,32,30);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,15,16,14)(17,21,22,18)(19,23,20,24)(25,30,31,27)(26,28,32,29)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=ghi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2109850);