Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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)(17,18,23,26,22,27,21,28)(19,20,29,32,30,24,25,31);(1,2)(3,7)(4,8)(5,6)(17,19,23,29,22,30,21,25)(18,24,26,31,27,20,28,32);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,24)(18,25,27,29)(19,28,30,26)(21,31,23,32);(1,3,5,4)(2,8,6,7)(17,21,22,23)(18,26,27,28)(19,29,30,25)(20,31,24,32);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(19,30)(20,24)(25,29)(31,32)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,e^b=ei,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290926);