Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(20,27)(24,29)(25,31)(30,32);(6,7)(10,11)(14,15)(18,22)(23,28)(24,29)(30,32);(6,7)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32);(1,2)(3,4)(5,7,8,6)(9,11,13,10)(12,15,16,14)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32);(9,12,13,16)(10,14,11,15)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)Polycyclic presentation:
2^9|a^2=fi,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836270);