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 | |
4498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(18,22)(19,24)(27,29)(32,33)(35,39)(36,40);(1,2)(3,7)(4,5)(6,8)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20)(25,26,31,34)(27,33,29,32)(28,37,38,30)(35,40,39,36);(2,5)(6,7)(13,21)(18,22)(19,24)(20,23)(25,27,31,29)(26,32,34,33)(28,35,38,39)(30,40,37,36);(3,8)(6,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,28)(26,30)(27,35)(29,39)(31,38)(32,40)(33,36)(34,37);(1,3,4,8)(2,6,5,7)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29,31,27)(26,33,34,32)(28,35,38,39)(30,40,37,36);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,18,15,22)(11,19,17,24)(12,20,16,23)(25,30)(26,28)(27,36)(29,40)(31,37)(32,39)(33,35)(34,38)Polycyclic presentation:
2^9|b^2=gh,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8268442);