Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10,12,14)(11,15,16,13)(18,23)(20,28)(21,31)(24,27);(1,3,5,7)(2,6,8,4)(9,11,12,16)(10,13,14,15)(18,23)(24,27)(25,30)(26,32);(3,7)(4,6)(17,18,22,23)(19,27,29,24)(20,30,28,25)(21,26,31,32);(17,19,22,29)(18,24,23,27)(20,21,28,31)(25,26,30,32);(1,4,5,6)(2,7,8,3)(17,20,22,28)(18,25,23,30)(19,21,29,31)(24,26,27,32);(1,2)(3,6)(4,7)(5,8)(17,21)(18,26)(19,28)(20,29)(22,31)(23,32)(24,30)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618158);