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 | |
4773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18,22,24)(19,27,28,23)(20,25,31,30)(21,32,29,26);(1,2)(3,7)(4,6)(5,8)(10,11)(14,15)(17,19,22,28)(18,23,24,27)(20,29,31,21)(25,32,30,26);(1,3)(2,5)(4,8)(6,7)(9,11,13,10)(12,15,16,14)(18,24)(23,27)(25,30)(26,32);(2,6)(5,7)(9,12,13,16)(10,14,11,15)(17,18)(19,27)(20,30)(21,26)(22,24)(23,28)(25,31)(29,32);(1,2)(3,5)(4,6)(7,8)(17,20,22,31)(18,25,24,30)(19,21,28,29)(23,26,27,32);(1,3)(2,7)(4,8)(5,6)(17,21)(18,26)(19,20)(22,29)(23,25)(24,32)(27,30)(28,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10126804);