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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,5,8,4)(11,12)(13,15)(18,21)(22,23)(25,26,29,31)(27,32,30,28);(1,3)(2,4)(5,7)(6,8)(9,10,14,16)(11,15,12,13)(17,18)(19,22)(20,21)(23,24)(25,27)(26,28)(29,30)(31,32);(1,4,6,5)(2,3,7,8)(9,11,14,12)(10,15,16,13)(17,18)(19,22)(20,21)(23,24)(25,26)(27,32)(28,30)(29,31);(1,5,6,4)(2,3,7,8)(9,12,14,11)(10,13,16,15)(17,19,20,24)(18,22,21,23)(25,28,29,32)(26,30,31,27);(1,3)(2,5)(4,7)(6,8)(9,13)(10,11)(12,16)(14,15)(17,18)(19,23)(20,21)(22,24)(25,26)(27,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=fg,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, 1212761);