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 | |
4451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,12)(13,21,23,16)(19,20,24,22)(25,26)(27,32)(28,30)(29,34)(31,33)(35,37)(36,39)(38,40);(1,2)(3,8)(4,6)(5,7)(10,15)(12,17)(16,21)(20,22)(25,27,31,28)(26,32,33,30)(29,36,38,35)(34,39,40,37);(1,3,5,4)(2,6,7,8)(10,15)(11,18)(16,21)(19,24)(26,33)(27,28)(34,40)(35,36);(1,2)(3,6)(4,8)(5,7)(9,11,14,18)(10,12,15,17)(13,19,23,24)(16,20,21,22)(25,28,31,27)(26,30,33,32)(29,35,38,36)(34,37,40,39);(1,4,5,3)(2,6,7,8)(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,29,31,38)(26,34,33,40)(27,35,28,36)(30,39,32,37);(1,2)(3,6)(4,8)(5,7)(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,30)(26,27)(28,33)(29,37)(31,32)(34,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=ghi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10063408);