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 | |
1683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,8,7,4)(10,11)(14,15)(17,18,22,24)(19,20,23,21);(2,6)(4,8)(9,10,13,11)(12,15,16,14)(17,19,22,23)(18,20,24,21)(26,29)(30,32);(2,6)(3,7)(10,11)(14,15)(17,20,22,21)(18,19,24,23)(25,26)(27,30)(28,29)(31,32);(1,3,5,7)(2,4,6,8)(9,11,13,10)(12,15,16,14)(17,18)(19,20)(21,23)(22,24);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,15)(13,16)(17,21,22,20)(18,23,24,19)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=fi,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, 1062683);