Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1940 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)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,15,12)(18,21)(19,23);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(18,21)(22,24)(25,26,28,30)(27,29,31,32);(9,11)(10,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,16)(13,15);(1,3)(2,7)(4,6)(5,8)(9,12,13,14)(10,15,16,11)(17,19)(18,22)(20,23)(21,24)(25,27,28,31)(26,29,30,32)Polycyclic presentation:
2^9|a^2=fi,b^2=fghi,d^2=i,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1223760);