Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
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)(19,23)(20,22)(21,24)(25,26)(27,30)(28,32)(29,31);(2,5)(3,8)(10,14)(12,15)(17,18,20,22)(19,23,24,21)(25,27,29,32)(26,30,31,28);(1,3,4,8)(2,6,5,7)(9,11,13,16)(10,15,14,12)(17,18,20,22)(19,21,24,23)(25,27)(26,28)(29,32)(30,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,28)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,21)(20,24)(22,23)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=fi,c^2=f,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, 1646236);