Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,21,22,25,20,30)(19,26,28,29,27,24,31,32);(1,2,4,5)(3,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,27)(18,24,25,26)(20,31,23,28)(21,32,30,29);(2,5)(6,8)(10,14)(13,16)(19,27)(24,26)(28,31)(29,32);(1,3,4,7)(2,6,5,8)(17,20,22,23)(18,21,25,30)(19,28,27,31)(24,29,26,32);(17,21,23,25,22,30,20,18)(19,29,28,24,27,32,31,26)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,d^b=di,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285948);