Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
2027 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)(3,8)(4,6)(5,7)(10,11)(12,16)(17,18,20,23)(19,22,24,21)(26,29)(30,32);(1,3,5,4)(2,6,7,8)(10,11)(14,15)(17,19,20,24)(18,21,23,22)(27,31)(30,32);(9,10,13,11)(12,14,16,15)(17,19)(18,21)(20,24)(22,23)(25,26)(27,30)(28,29)(31,32);(1,4,5,3)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,24)(21,23)(25,27)(26,30)(28,31)(29,32);(1,2)(3,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,23)(22,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=i,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2542739);