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 | |
4449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,19,22)(12,18,23,20)(13,21,24,14)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,19)(10,16,17,22)(12,21,23,14)(13,18,24,20)(26,32)(27,28)(34,37)(35,36);(2,5)(6,7)(10,17)(11,19)(14,21)(18,20)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(3,8)(6,7)(9,12)(10,18)(11,14)(13,16)(15,23)(17,20)(19,21)(22,24)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,3,4,8)(2,6,5,7)(9,13)(10,14)(11,20)(12,22)(15,24)(16,23)(17,21)(18,19)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,13)(11,12)(15,21)(16,18)(17,24)(19,23)(20,22)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=ghi,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, 8268969);