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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1729 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)(12,14)(13,16)(18,21)(22,24)(26,30)(28,31);(9,10,12,14)(11,13,16,15)(19,23)(22,24)(26,30)(27,32);(11,16)(13,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,30);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,20,23)(18,22,21,24)(25,27,29,32)(26,31,30,28);(1,3,4,7)(2,5,6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,28,29,31)(26,32,30,27)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=g,e^2=fhi,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, 1746726);