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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2058 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)(9,10,12,14)(11,13,15,16)(18,21)(19,23)(26,30)(27,31);(2,3)(4,8)(18,21)(22,24)(25,26)(27,32)(28,31)(29,30);(2,3)(6,7)(17,18)(19,22)(20,21)(23,24)(25,26,29,30)(27,32,31,28);(1,3,5,2)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,27,29,31)(26,28,30,32);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,13,14,16)(17,19,20,23)(18,22,21,24)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=gi,c^2=fgh,d^2=f,e^2=fghi,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, 3136101);