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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,5,4)(9,10,15,20)(11,14,19,16)(12,17,22,24)(13,18,21,23)(26,32)(27,37)(28,30)(35,40);(1,3,6,5)(2,4,8,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,18)(23,24)(25,26)(27,38)(28,34)(29,40)(30,36)(31,32)(33,37)(35,39);(1,4)(2,5)(3,8)(6,7)(9,12,15,22)(10,17,20,24)(11,21,19,13)(14,23,16,18)(25,27)(26,33)(28,29)(30,39)(31,37)(32,38)(34,35)(36,40);(1,4,6,7)(2,3,8,5)(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,18)(20,24)(25,28,31,30)(26,34,32,36)(27,29,37,39)(33,35,38,40);(1,4)(2,3)(5,8)(6,7)(9,13,15,21)(10,18,20,23)(11,22,19,12)(14,24,16,17)(25,29,31,39)(26,35,32,40)(27,28,37,30)(33,34,38,36);(1,5,6,3)(2,7,8,4)(9,14)(10,19)(11,20)(12,23)(13,17)(15,16)(18,22)(21,24)(25,30,31,28)(26,36,32,34)(27,39,37,29)(33,40,38,35)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=gh,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113836);