Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1574 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)(12,16)(13,15)(17,18,23,26,22,27,21,30)(19,25,28,31,20,24,29,32);(1,2,5,6)(3,7,4,8)(9,11,14,10)(12,16,13,15)(17,19,22,20)(18,24,27,25)(21,29,23,28)(26,32,30,31);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16)(17,20,22,19)(18,25,27,24)(21,28,23,29)(26,31,30,32);(1,3,5,4)(2,8,6,7)(9,13,14,12)(10,15,11,16)(17,21,22,23)(18,26,27,30)(19,28,20,29)(24,31,25,32);(1,4,5,3)(2,8,6,7)(19,20)(24,25)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,e^b=ei,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, 6288527);