Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1373 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)(11,14)(12,13)(15,16)(17,18,23,19,22,25,21,27)(20,28,29,32,30,24,31,26);(1,2)(3,7)(4,8)(5,6)(17,19,23,25,22,27,21,18)(20,26,29,28,30,32,31,24);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,30)(18,24,25,28)(19,26,27,32)(21,31,23,29);(1,3,5,4)(2,8,6,7)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,32,28,26);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(20,30)(24,28)(26,32)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,d^c=di,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290925);