Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,24)(21,25)(23,28)(26,30)(27,31)(29,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,21,28)(18,22,25,23)(20,26,29,27)(24,30,32,31);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(19,26)(20,29)(22,30)(23,31)(24,32)(27,28);(17,19,21,28)(18,23,25,22)(20,26,29,27)(24,31,32,30);(19,26)(20,29)(22,30)(23,31)(24,32)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486696);