Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| 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 | |
1844 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)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(17,18,22,24)(19,20,23,21)(25,26,29,28)(27,31,32,30);(2,6)(7,8)(9,11)(10,14)(12,13)(15,16)(17,19,22,23)(18,20,24,21)(26,28)(30,31);(1,3,5,4)(2,7,6,8)(9,10)(11,16)(12,14)(13,15)(17,20,22,21)(18,19,24,23)(25,26,29,28)(27,30,32,31);(1,4,5,3)(2,7,6,8)(9,10)(11,14)(12,16)(13,15)(17,21,22,20)(18,23,24,19)(25,27)(26,30)(28,31)(29,32);(1,2)(3,7)(4,8)(5,6)(9,12,13,11)(10,14,15,16)(17,18)(19,20)(21,23)(22,24)(25,28,29,26)(27,30,32,31)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=i,d^2=fgh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1398572);