Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806527 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1380 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,23,22)(19,24,28,31)(20,21,25,29)(26,27,30,32);(9,11)(10,13)(12,16)(14,15)(17,19,23,28)(18,24,22,31)(20,27,25,32)(21,30,29,26);(2,6)(7,8)(10,14)(11,16)(17,20,22,29,23,25,18,21)(19,26,24,27,28,30,31,32);(1,3,5,4)(2,7,6,8)(10,14)(13,15);(1,4,5,3)(2,8,6,7)(17,21,18,25,23,29,22,20)(19,27,31,26,28,32,24,30)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=fg,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6748874);