Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,24)(19,20,21,23)(25,26,29,31)(27,28,30,32);(9,10,13,12)(11,14,16,15)(17,19)(18,23)(20,24)(21,22)(25,27)(26,28)(29,30)(31,32);(10,12)(14,15)(17,20,22,23)(18,21,24,19)(25,28)(26,30)(27,31)(29,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,19)(18,20,24,23);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,14,16,15)(17,20)(18,21)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=fg,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3537894);