Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 6291456 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4374 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)(5,6,8,7)(10,13)(14,16)(19,25)(20,26)(21,29)(23,30);(6,7)(17,18,22,24)(19,26,25,20)(21,23,29,30)(27,32,31,28);(2,4)(6,7)(17,19,22,25)(18,20,24,26)(21,27,29,31)(23,28,30,32);(1,2)(3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,28)(22,26)(23,27)(24,25)(29,32)(30,31);(9,11,12,15)(10,14,13,16)(17,21)(18,23)(19,27)(20,28)(22,29)(24,30)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,d^2=f,e^2=hi,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, 4839541);