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 | |
4319 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)(9,10,14,16)(11,18,21,15)(12,13,23,17)(19,20,22,24)(25,26,31,35)(27,36,37,32)(28,38,39,33)(29,30,34,40);(2,6)(3,7)(9,11,14,21)(10,15,16,18)(12,22,23,19)(13,20,17,24)(25,27,31,37)(26,32,35,36)(28,34,39,29)(30,33,40,38);(1,3)(2,4)(5,7)(6,8)(10,16)(12,23)(15,18)(19,22);(1,4)(2,3)(5,8)(6,7)(9,12)(10,17)(11,19)(13,16)(14,23)(15,20)(18,24)(21,22)(25,28,31,39)(26,33,35,38)(27,29,37,34)(30,36,40,32);(9,11)(10,18)(12,22)(13,24)(14,21)(15,16)(17,20)(19,23)(25,29)(26,30)(27,28)(31,34)(32,33)(35,40)(36,38)(37,39);(9,13,14,17)(10,12,16,23)(11,20,21,24)(15,22,18,19)(25,30)(26,34)(27,33)(28,36)(29,35)(31,40)(32,39)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=ghi,f^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, 10154849);