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 | |
4463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,15,19)(11,20,21,16)(12,22,23,17)(13,14,18,24)(25,26,31,34)(27,28,29,32)(30,38,39,33)(35,40,37,36);(2,6)(3,7)(9,11,15,21)(10,16,19,20)(12,18,23,13)(14,17,24,22)(25,27)(26,32)(28,34)(29,31)(30,35)(33,40)(36,38)(37,39);(1,3)(2,4)(5,7)(6,8)(25,28,31,32)(26,27,34,29)(30,36,39,40)(33,35,38,37);(1,4)(2,3)(5,8)(6,7)(9,12,15,23)(10,17,19,22)(11,13,21,18)(14,20,24,16)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,38)(39,40);(9,13)(10,14)(11,12)(15,18)(16,17)(19,24)(20,22)(21,23)(25,29,31,27)(26,32,34,28)(30,35,39,37)(33,36,38,40);(9,14)(10,18)(11,17)(12,20)(13,19)(15,24)(16,23)(21,22)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=ghi,e^2=h,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, 10154486);