Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1752 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)(3,5)(4,6)(7,8)(9,10,14,16)(11,12,15,13)(18,22)(19,23);(1,3)(2,7)(4,5)(6,8)(9,11,14,15)(10,12,16,13)(17,18)(19,24)(20,23)(21,22)(26,29)(30,32);(2,4)(3,8)(9,12,14,13)(10,11,16,15)(17,19,21,23)(18,20,22,24)(25,26)(27,30)(28,29)(31,32);(1,4,6,2)(3,5,8,7)(9,10)(11,12)(13,15)(14,16)(17,20,21,24)(18,23,22,19);(1,5)(2,8)(3,4)(6,7)(9,13,14,12)(10,15,16,11)(17,18)(19,20)(21,22)(23,24)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fi,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1313460);