Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(9,10,12,13)(11,15,16,14)(17,18,21,24)(19,20,23,22)(25,26)(27,28)(29,30)(31,32);(1,2,5,3)(4,6,8,7)(10,13)(14,15)(17,19)(18,22)(20,24)(21,23)(26,30)(28,31);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,30)(29,32);(1,3,5,2)(4,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,24)(22,23)(25,28,29,31)(26,27,30,32);(1,4)(2,6)(3,7)(5,8)(9,11,12,16)(10,14,13,15)(17,20)(18,23)(19,24)(21,22)(25,27)(26,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4802856);