Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | #15763 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2312 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)(6,8)(9,10,13,14)(11,12,16,15)(18,21)(22,24)(26,30)(27,32);(10,14)(11,16)(17,18,20,21)(19,22,23,24)(25,26,29,30)(27,28,32,31);(10,14)(12,15)(17,19,20,23)(18,22,21,24)(25,27,29,32)(26,31,30,28);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,15,14,12)(17,18)(19,22)(20,21)(23,24)(25,28)(26,27)(29,31)(30,32);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,16,14,11)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,c^2=fgh,d^2=gh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4317172);