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 | |
1842 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,22)(19,20,24,23)(27,31)(30,32);(1,3,6,5)(2,7,4,8)(9,10,14,13)(11,16,12,15)(18,22)(20,23)(26,29)(30,32);(1,2,6,4)(3,8,5,7)(9,11,14,12)(10,15,13,16)(17,19)(18,23)(20,22)(21,24)(25,26)(27,30)(28,29)(31,32);(1,4,6,2)(3,7,5,8)(9,12,14,11)(10,15,13,16)(17,20,21,23)(18,19,22,24);(1,5,6,3)(2,7,4,8)(9,13,14,10)(11,16,12,15)(17,19)(18,20)(21,24)(22,23)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=fg,d^2=i,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, 4802835);