Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2076 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)(3,4)(5,8)(6,7)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,22)(21,24)(27,31)(30,32);(1,3,6,5)(2,4,7,8)(10,12)(14,15)(17,19,22,21)(18,23,20,24)(26,29)(30,32);(2,7)(4,8)(9,10,13,12)(11,15,16,14)(17,19,22,21)(18,24,20,23)(25,26)(27,30)(28,29)(31,32);(1,4,6,8)(2,3,7,5)(9,11,13,16)(10,14,12,15)(17,20,22,18)(19,24,21,23);(1,5,6,3)(2,8,7,4)(9,12,13,10)(11,15,16,14)(17,21,22,19)(18,24,20,23)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4869739);