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 | #14757 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2367 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)(10,14)(11,15)(18,21)(22,24)(25,26)(27,32)(28,31)(29,30);(2,5)(3,7)(9,10,13,14)(11,16,15,12)(19,23)(22,24)(26,30)(28,32);(1,2,4,5)(3,6,7,8)(9,11,13,15)(10,12,14,16)(17,18)(19,22)(20,21)(23,24)(25,27,29,31)(26,28,30,32);(1,2)(3,8)(4,5)(6,7)(9,11)(10,12)(13,15)(14,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,32)(26,31,30,27);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=fh,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, 2997109);