Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996315 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(10,14)(12,15)(17,18,22,25)(19,23,21,30)(20,27,29,32)(24,26,31,28);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,12,16,15)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,27,31,32);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,20,19,26,22,29,21,28)(18,24,30,32,25,31,23,27);(3,8)(6,7)(10,14)(11,16)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,27,31,32);(9,12,13,15)(10,11,14,16)Polycyclic presentation:
2^9|a^2=fhi,b^2=hi,c^2=fghi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105482);