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 | |
4504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26,30,34)(27,35,37,31)(28,38,39,32)(29,33,36,40);(1,3)(2,6)(4,7)(5,8)(11,17)(14,21)(18,23)(22,24)(25,27,30,37)(26,31,34,35)(28,29,39,36)(32,33,38,40);(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,28,30,39)(26,32,34,38)(27,36,37,29)(31,40,35,33);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28)(26,32)(27,29)(30,39)(31,33)(34,38)(35,40)(36,37);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,29)(26,33)(27,28)(30,36)(31,32)(34,40)(35,38)(37,39);(1,4,5,2)(3,7,8,6)(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154718);