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 | #37714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,16)(12,17)(18,24)(20,22)(26,32)(27,35)(28,36)(29,39);(1,2)(3,6)(4,7)(5,8)(9,10,15,16)(11,17,13,12)(14,18,23,24)(19,22,21,20)(25,26)(27,33)(28,37)(29,30)(31,32)(34,36)(35,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,13)(10,17,16,12)(14,19,23,21)(18,22,24,20)(25,27,31,35)(26,33,32,38)(28,30,36,40)(29,34,39,37);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,22)(15,17)(18,21)(19,24)(20,23)(25,28)(26,34)(27,30)(29,38)(31,36)(32,37)(33,39)(35,40);(9,13,15,11)(10,12,16,17)(14,21,23,19)(18,20,24,22)(25,29)(26,30)(27,34)(28,38)(31,39)(32,40)(33,36)(35,37);(1,2)(3,5)(4,7)(6,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,24)(17,22)(25,30)(26,29)(27,36)(28,35)(31,40)(32,39)(33,34)(37,38)Polycyclic presentation:
2^9|b^2=gh,c^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8329274);