Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,13,14)(11,15,16,12)(17,18)(19,21)(20,22)(23,24);(2,5)(3,8)(10,14)(12,15)(17,18,20,22)(19,21,23,24)(25,26,28,30)(27,29,31,32);(2,5)(6,7)(9,11,13,16)(10,15,14,12)(17,19,20,23)(18,21,22,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,21)(20,23)(22,24);(1,3,4,8)(2,6,5,7)(9,12)(10,11)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,27,28,31)(26,29,30,32)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1731071);