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 | |
4871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(3,7)(10,12)(14,15)(17,18,22,25)(19,23,26,20)(21,24,30,32)(27,31,29,28);(2,5)(3,7)(9,10,13,12)(11,15,16,14)(17,19,22,26)(18,23,25,20)(21,29,30,27)(24,28,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(19,26)(20,23)(27,29)(28,31);(2,5)(6,8)(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,29)(30,32);(1,3,4,7)(2,6,5,8)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8735870);