Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,6,4,7)(9,10,14,16)(11,13,15,12)(18,21)(19,23)(26,30)(28,32);(1,3,5,4)(2,6,8,7)(9,11,14,15)(10,12,16,13)(19,23)(22,24)(26,30)(27,31);(9,12,14,13)(10,15,16,11)(17,18)(19,22)(20,21)(23,24)(25,26,29,30)(27,28,31,32);(1,2)(3,6)(4,7)(5,8)(9,13,14,12)(10,15,16,11)(17,19)(18,22)(20,23)(21,24)(25,27,29,31)(26,28,30,32);(1,4,5,3)(2,7,8,6)(9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|a^2=fghi,b^2=fghi,c^2=h,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2434156);