Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,4,8)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,30)(21,23)(22,26)(25,27)(28,32)(29,31);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(18,23)(19,27)(20,29)(22,28)(24,30)(25,31);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,19,28,20,21,29,22,27)(18,24,32,30,23,31,26,25);(10,14)(13,15)(18,23)(24,31)(25,30)(26,32);(17,20,22,19,21,27,28,29)(18,25,26,31,23,30,32,24)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=ghi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6727990);