Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2548 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)(3,7)(10,11)(14,15)(18,21)(19,23)(25,26)(27,28)(29,30)(31,32);(2,5)(6,8)(9,10)(11,13)(12,15)(14,16)(25,27)(26,30)(28,32)(29,31);(10,11)(14,15)(17,18,20,21)(19,22,23,24)(25,28,31,30)(26,27,32,29);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,29,31,27)(26,28,32,30);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,19)(18,22)(20,23)(21,24)(25,30,31,28)(26,27,32,29)Polycyclic presentation:
2^9|c^2=fgh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5867743);