Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,21)(20,22)(23,24)(26,28)(27,31);(10,14)(11,16)(19,23)(21,24)(26,28)(30,32);(11,16)(12,15)(17,18,20,22)(19,21,23,24)(25,26,29,28)(27,32,31,30);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,15,14,12)(17,18)(19,21)(20,22)(23,24)(25,27,29,31)(26,30,28,32);(1,3,4,7)(2,5,6,8)(9,12,13,15)(10,16,14,11)(17,19,20,23)(18,21,22,24)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=hi,c^2=g,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5090147);