Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,26,28,30)(27,29,31,32);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(18,22)(19,24)(25,27,28,31)(26,29,30,32);(9,10,12,15)(11,14,16,13)(17,19,21,24)(18,23,22,20)(25,27)(26,29)(28,31)(30,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,20,21,23)(18,24,22,19);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fghi,b^2=gh,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5897700);