Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| 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 | #14757 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,8,4,7)(10,11)(12,16)(17,18,20,22)(19,23,24,21)(25,26)(27,28)(29,30)(31,32);(2,6)(7,8)(9,10,13,11)(12,15,16,14)(26,30)(28,32);(1,3,5,4)(2,7,6,8)(9,10,13,11)(12,14,16,15)(17,18)(19,21)(20,22)(23,24)(25,26,29,30)(27,28,31,32);(1,4,5,3)(2,7,6,8)(9,11,13,10)(12,14,16,15)(17,19,20,24)(18,21,22,23)(25,27)(26,28)(29,31)(30,32);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=fgi,c^2=f,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1359236);