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 | |
4484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,6,8,7)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,2)(3,5)(4,6)(7,8)(9,10,14,15)(11,17,20,16)(12,21,23,13)(18,24,22,19)(26,32)(27,28)(34,37)(35,36);(4,8)(6,7)(10,15)(13,21)(16,17)(19,24)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(2,3)(6,7)(9,11,14,20)(10,16,15,17)(12,22,23,18)(13,19,21,24)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,3,5,2)(4,7,8,6)(9,12)(10,13)(11,18)(14,23)(15,21)(16,24)(17,19)(20,22)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(1,4,5,8)(2,6,3,7)(9,13)(10,12)(11,19)(14,21)(15,23)(16,22)(17,18)(20,24)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300498);