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 | |
4540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,16,20)(11,19,14,15)(12,18,13,17)(21,22,23,24)(25,26,32,35)(27,36,38,33)(28,31,29,39)(30,34,40,37);(1,2,4,5)(3,6,7,8)(9,11,16,14)(10,15,20,19)(12,21,13,23)(17,22,18,24)(25,27)(26,33)(28,34)(29,37)(30,31)(32,38)(35,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,17,20,18)(11,21,14,23)(15,22,19,24)(25,28,32,29)(26,31,35,39)(27,34,38,37)(30,36,40,33);(1,2)(3,6)(4,5)(7,8)(9,13,16,12)(10,18,20,17)(11,21,14,23)(15,22,19,24)(25,29,32,28)(26,31,35,39)(27,37,38,34)(30,36,40,33);(1,3)(2,6)(4,7)(5,8)(9,14,16,11)(10,19,20,15)(12,21,13,23)(17,22,18,24)(25,30)(26,34)(27,31)(28,33)(29,36)(32,40)(35,37)(38,39);(9,15,16,19)(10,11,20,14)(12,22,13,24)(17,21,18,23)(25,31)(26,29)(27,30)(28,35)(32,39)(33,37)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gi,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8488973);