Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1982 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,23,24,21)(25,26,29,30)(27,31,32,28);(2,5)(3,7)(9,11,14,16)(10,15,12,13)(17,18,20,22)(19,21,24,23)(27,32)(28,31);(2,5)(6,8)(10,12)(11,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,28)(29,32)(30,31);(1,2)(3,8)(4,5)(6,7)(9,12,14,10)(11,15,16,13)(17,18)(19,21)(20,22)(23,24)(25,28)(26,27)(29,31)(30,32);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,11,12,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2997410);