Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | |
2082 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)(6,7)(9,10)(11,16)(12,14)(13,15)(19,20)(21,23)(25,26,28,30)(27,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,18,22,24)(19,23,20,21)(27,32)(29,31);(17,19,22,20)(18,23,24,21)(25,27,28,32)(26,29,30,31);(1,2)(3,7)(4,5)(6,8)(9,12,13,11)(10,14,15,16)(17,20,22,19)(18,21,24,23)(25,26)(27,29)(28,30)(31,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,21)(18,19)(20,24)(22,23)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=g,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1212759);