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 | |
4541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,11)(12,19)(16,18)(20,21)(25,26)(27,32)(28,34)(29,38)(30,36)(31,37)(33,39)(35,40);(1,2)(3,7)(4,6)(5,8)(9,10,14,11)(12,15,19,17)(13,16,22,18)(20,23,21,24)(25,27,31,30)(26,32,37,36)(28,35,39,38)(29,34,40,33);(1,3)(2,5)(4,8)(6,7)(9,10,14,11)(12,15,19,17)(13,16,22,18)(20,23,21,24)(25,28)(26,33)(27,38)(29,36)(30,35)(31,39)(32,40)(34,37);(2,6)(5,7)(9,11,14,10)(12,17,19,15)(13,16,22,18)(20,23,21,24)(25,28)(26,34)(27,35)(29,36)(30,38)(31,39)(32,40)(33,37);(1,3)(2,7)(4,8)(5,6)(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,29)(26,35)(27,34)(28,36)(30,33)(31,40)(32,39)(37,38);(1,2)(3,5)(4,6)(7,8)(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,30,31,27)(26,36,37,32)(28,35,39,38)(29,34,40,33)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8597177);