Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,13,16)(11,15,12,14);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,18)(19,26)(20,25)(21,29)(22,24)(23,27)(28,32)(30,31);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,15,16,14)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(1,4,5,3)(2,6,8,7)(18,24)(23,31)(25,29)(26,32);(17,20,22,21)(18,25,24,29)(19,27,28,30)(23,32,31,26)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,h^b=hi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339893);