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 | |
1400 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,21,22,26,28,32)(19,27,31,30,29,24,20,25);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,28)(21,25)(22,29)(23,31)(26,27)(30,32);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20,22,31)(18,25,26,30)(19,28,29,23)(21,27,32,24);(2,5)(6,8)(18,26)(19,29)(20,31)(21,32);(17,21,23,26,22,32,28,18)(19,30,31,24,29,25,20,27)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cfi,d^b=di,e^b=ei,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285674);