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 | |
4986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(6,8)(9,10,13,11)(12,15,16,14)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32);(1,2,4,5)(3,6,7,8)(10,11)(14,15)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(18,25)(19,28)(20,30)(26,32);(2,5)(6,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31);(1,2)(3,8)(4,5)(6,7)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30);(1,3)(2,6)(4,7)(5,8)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,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, 10219856);