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:
(2,6)(3,7)(10,11)(12,16)(18,21)(22,24)(25,26,29,31)(27,32,30,28);(1,2,5,6)(3,4,7,8)(10,11)(14,15)(19,23)(22,24)(25,27,29,30)(26,28,31,32);(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,32)(28,30)(29,31);(1,3)(2,4)(5,7)(6,8)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,23)(21,24)(25,28,29,32)(26,30,31,27);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=fi,d^2=h,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, 2542736);