Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | #17086 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(18,22)(19,24)(25,26)(27,31)(28,32)(29,30);(1,2)(3,8)(4,5)(6,7)(9,10,13,11)(12,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,28)(29,32)(30,31);(3,7)(6,8)(10,11)(12,16)(18,22)(20,23)(26,30)(28,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,22)(21,24)(25,26)(27,28)(29,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,20,21,23)(18,19,22,24)(25,28)(26,27)(29,31)(30,32)Polycyclic presentation:
2^9|b^2=fgi,d^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5426900);