Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(10,11)(12,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,29)(30,32);(4,8)(6,7)(9,10,13,11)(12,15,16,14)(17,18,21,19)(20,22,24,23)(25,27,30,28)(26,31,32,29);(2,3)(6,7)(12,16)(14,15)(18,19)(20,24)(26,32)(27,28);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,19,21,18)(20,23,24,22)(25,28,30,27)(26,31,32,29);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20,21,24)(18,22,19,23)(25,29,30,31)(26,27,32,28)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5429483);