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 |
| prime radical | #45917 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,13)(14,16)(17,21)(18,19)(20,22)(23,24)(25,26,31,35)(27,36,38,32)(28,30,34,40)(29,33,39,37);(1,2)(3,7)(4,6)(5,8)(9,11,14,20)(10,15,16,22)(12,18,17,23)(13,19,21,24)(25,27,31,38)(26,32,35,36)(28,29,34,39)(30,37,40,33);(1,3)(2,5)(4,8)(6,7)(10,16)(12,17)(15,22)(18,23)(25,28,31,34)(26,30,35,40)(27,29,38,39)(32,37,36,33);(2,6)(5,7)(9,12,14,17)(10,13,16,21)(11,18,20,23)(15,19,22,24)(25,27)(26,32)(28,29)(30,37)(31,38)(33,40)(34,39)(35,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,29)(26,33)(27,34)(28,38)(30,32)(31,39)(35,37)(36,40);(1,3)(2,7)(4,8)(5,6)(9,13)(10,17)(11,19)(12,16)(14,21)(15,23)(18,22)(20,24)(25,30)(26,34)(27,37)(28,35)(29,36)(31,40)(32,39)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10125542);