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 | |
1976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,14,15)(11,16,13,12)(17,18,21,22)(19,24,23,20)(25,26,29,31)(27,30,32,28);(2,5)(6,8)(9,11,14,13)(10,12,15,16)(17,19)(18,20)(21,23)(22,24);(2,5)(6,8)(10,15)(12,16)(18,22)(19,23)(25,27,29,32)(26,28,31,30);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,24)(20,23)(21,22)(25,27)(26,30)(28,31)(29,32);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,12,15,16)(17,20,21,24)(18,23,22,19)(25,28,29,30)(26,27,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=fg,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, 4836552);