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 | |
4638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10,15,14)(11,18,13,16)(12,20,21,17)(19,23,22,24)(25,26)(27,31)(28,36)(29,33)(30,34)(32,37)(35,38)(39,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,14,18)(12,19,21,22)(17,23,20,24)(27,34)(30,31)(35,40)(38,39);(1,2)(3,4)(5,6)(7,8)(12,21)(17,20)(19,22)(23,24)(25,27,29,34)(26,30,33,31)(28,35,37,40)(32,38,36,39);(1,3,5,8)(2,7,6,4)(9,12)(10,17)(11,19)(13,22)(14,20)(15,21)(16,23)(18,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,2)(3,7)(4,8)(5,6)(9,13,15,11)(10,16,14,18)(12,22,21,19)(17,23,20,24)(25,28,29,37)(26,32,33,36)(27,35,34,40)(30,38,31,39);(1,4,5,7)(2,3,6,8)(9,14,15,10)(11,18,13,16)(12,17,21,20)(19,23,22,24)(25,26)(27,30)(28,36)(29,33)(31,34)(32,37)(35,39)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627790);