Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1984 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)(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,19)(20,23,24,22)(25,26,30,31)(27,28,29,32);(9,11,13,12)(10,14,15,16)(18,19)(22,23)(26,31)(28,32);(2,6)(3,8)(10,15)(14,16)(17,18,21,19)(20,22,24,23)(25,27,30,29)(26,32,31,28);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19,21,18)(20,23,24,22)(25,28,30,32)(26,27,31,29);(1,4,5,7)(2,3,6,8)(9,12,13,11)(10,16,15,14)(17,20)(18,22)(19,23)(21,24)(25,29,30,27)(26,32,31,28)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836535);