Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| 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 | |
1581 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)(3,8)(4,7)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,20,23)(19,22,24,21)(27,31)(29,32);(1,3,6,4)(2,7,5,8)(10,11)(12,16)(17,19,20,24)(18,21,23,22);(9,10,13,11)(12,14,16,15)(17,19)(18,22)(20,24)(21,23)(25,26,28,30)(27,29,31,32);(1,4,6,3)(2,7,5,8)(9,11,13,10)(12,15,16,14)(25,27,28,31)(26,29,30,32);(1,5,6,2)(3,7,4,8)(9,12,13,16)(10,14,11,15)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=i,d^2=fhi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1980341);