Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2902 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)(3,5,7,8)(9,10,13,16)(11,12,15,14)(17,18,20,21)(19,23,24,22)(26,28)(27,31);(9,11)(10,14)(12,16)(13,15)(18,21)(19,24)(26,28)(30,32);(11,15)(12,14)(17,18)(19,23)(20,21)(22,24)(25,26,29,28)(27,30,31,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,23)(20,21)(22,24)(25,27,29,31)(26,30,28,32);(1,3,4,7)(2,5,6,8)(9,12)(10,15)(11,16)(13,14)(17,19,20,24)(18,22,21,23)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=fhi,c^2=fh,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4028740);