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 | |
4784 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,19)(13,16)(15,20)(17,23)(18,22)(21,24)(26,31)(27,34)(29,37)(36,39);(1,3,5,4)(2,6,7,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(9,10,13,16)(11,14,18,22)(12,15,20,19)(17,21,24,23)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,39)(38,40);(1,4,5,3)(2,6,7,8)(9,12,13,20)(10,15,16,19)(11,17,18,24)(14,21,22,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|b^2=ghi,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, 10154792);