Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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 | #59033 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,29)(21,23)(22,28)(24,27)(26,31)(30,32);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(19,26)(22,30)(25,31)(28,32);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,27,32,21,26,20,28)(18,22,29,31,23,30,24,25);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(19,26)(22,30)(25,31)(28,32);(10,14)(13,15)(18,23)(19,26)(24,29)(28,32)Polycyclic presentation:
2^9|b^2=f,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6931343);