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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,7)(3,6,8,5)(11,15)(14,16)(19,21)(20,23);(1,3)(2,5)(4,8)(6,7)(17,18,22,24)(19,23,21,20)(25,26,29,28)(27,31,32,30);(1,3)(2,5)(4,8)(6,7)(9,10,13,12)(11,14,15,16)(17,19,22,21)(18,23,24,20)(25,27,29,32)(26,30,28,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,24)(22,23)(25,26,29,28)(27,30,32,31);(1,2)(3,5)(4,7)(6,8)(9,12,13,10)(11,16,15,14)(17,21,22,19)(18,20,24,23)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=hi,d^2=h,e^2=hi,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, 1299344);