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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,15)(12,13)(14,16)(18,19)(20,24)(25,26,30,32)(27,29,31,28);(1,3,6,5)(2,7,4,8)(9,10,13,12)(11,14,16,15)(18,19)(22,23)(25,27,30,31)(26,28,32,29);(3,5)(7,8)(10,12)(14,15)(17,18,21,19)(20,22,24,23)(25,28,30,29)(26,31,32,27);(1,4,6,2)(3,7,5,8)(9,11,13,16)(10,14,12,15)(17,19,21,18)(20,23,24,22)(25,29,30,28)(26,31,32,27);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,15,16,14)(17,20,21,24)(18,22,19,23)(25,26)(27,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=fghi,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, 2456223);