Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2145 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)(3,6)(4,7)(5,8)(10,14)(11,15)(19,24)(20,23)(27,31)(30,32);(1,3)(2,5)(4,8)(6,7)(9,10,13,14)(11,12,15,16)(17,18,22,21)(19,23,24,20)(26,29)(30,32);(17,19,22,24)(18,23,21,20)(25,26,28,29)(27,30,31,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23);(1,2)(3,5)(4,7)(6,8)(9,12)(10,15)(11,14)(13,16)(17,21,22,18)(19,20,24,23)(25,27,28,31)(26,30,29,32)Polycyclic presentation:
2^9|b^2=i,c^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4336075);