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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,16)(12,15)(13,14)(17,18,23,21)(19,24,22,20)(25,26)(27,31)(28,29)(30,32);(1,2,4,5)(3,6,7,8)(9,11,14,12)(10,15,13,16)(17,19,23,22)(18,24,21,20)(26,29)(27,32);(9,11,14,12)(10,16,13,15)(17,20)(18,22)(19,21)(23,24)(25,26)(27,30)(28,29)(31,32);(1,3)(2,6)(4,7)(5,8)(9,12,14,11)(10,15,13,16)(17,21,23,18)(19,24,22,20);(1,2)(3,6)(4,5)(7,8)(9,13,14,10)(11,15,12,16)(17,22,23,19)(18,24,21,20)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fghi,d^2=fhi,e^2=fhi,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, 1983680);