Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1612 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,21,23,20,25)(19,27,28,32,29,24,30,31);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(18,23)(24,27)(25,26)(31,32);(10,14)(13,16)(17,19,22,28,21,29,20,30)(18,24,26,31,23,27,25,32);(1,4,6,5)(2,7,3,8)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,31,27,32);(1,5,6,4)(2,7,3,8)(17,20,21,22)(18,26,23,25)(19,28,29,30)(24,32,27,31)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300511);