Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15123 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(10,12)(11,15)(17,18)(19,20)(21,22)(23,24)(25,26,29,30)(27,31,32,28);(2,6)(3,8)(9,10,13,12)(11,14,15,16)(17,19)(18,23)(20,21)(22,24)(26,30)(27,32);(2,6)(4,7)(10,12)(14,16)(18,21)(19,24)(25,27)(26,31)(28,30)(29,32);(1,3,5,8)(2,7,6,4)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,24)(22,23);(1,4,5,7)(2,8,6,3)(9,12,13,10)(11,16,15,14)(17,21,22,18)(19,20,24,23)(25,28,29,31)(26,32,30,27)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 937431);