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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4656 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)(3,6)(4,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,24,27)(19,22,26,32)(20,29,31,25)(21,23,30,28);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(17,19,24,26)(18,22,27,32)(20,28,31,23)(21,25,30,29);(9,10)(11,14)(12,13)(15,16)(17,20,24,31)(18,25,27,29)(19,28,26,23)(21,22,30,32);(1,2)(3,6)(4,7)(5,8)(9,11,12,16)(10,14,13,15)(17,21)(18,23)(19,25)(20,22)(24,30)(26,29)(27,28)(31,32);(1,2)(3,5)(4,7)(6,8)(17,22)(18,26)(19,27)(20,30)(21,31)(23,25)(24,32)(28,29);(1,3)(2,6)(4,8)(5,7)(17,23)(18,21)(19,20)(22,25)(24,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10388219);