Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(10,12)(11,15)(17,18,21,20)(19,22,23,24)(25,26,28,30)(27,31,32,29);(1,3,6,4)(2,7,8,5)(19,23)(22,24)(25,26,28,30)(27,29,32,31);(2,8)(3,4)(9,10,13,12)(11,14,15,16)(17,19,21,23)(18,22,20,24)(25,26)(27,29)(28,30)(31,32);(1,4,6,3)(2,7,8,5)(9,11)(10,14)(12,16)(13,15);(1,5,6,7)(2,3,8,4)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,24,23,22)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1033789);