Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14757 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(10,14)(11,12)(17,18,22,24)(19,21,23,20)(25,26,29,30)(27,28,31,32);(1,3,6,4)(2,7,8,5)(9,10,13,14)(11,15,12,16)(17,19)(18,20)(21,24)(22,23)(26,30)(28,32);(1,3,6,4)(2,5,8,7)(9,11,13,12)(10,15,14,16)(17,20,22,21)(18,23,24,19)(25,26,29,30)(27,28,31,32);(1,4,6,3)(2,5,8,7)(9,12,13,11)(10,15,14,16)(17,21,22,20)(18,23,24,19)(25,27)(26,28)(29,31)(30,32);(1,5)(2,4)(3,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,21)(20,23)(22,24)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=f,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2751317);