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 | |
4631 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)(11,13)(12,19)(14,22)(15,17)(16,24)(18,20)(21,23)(25,26,30,34)(27,35,38,31)(28,32,37,39)(29,40,36,33);(1,2)(3,4)(5,7)(6,8)(9,11,15,19)(10,13,17,12)(14,18,24,23)(16,21,22,20)(25,27,30,38)(26,31,34,35)(28,29,37,36)(32,33,39,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,13)(10,11,17,19)(14,20,24,21)(16,18,22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,40)(37,39);(1,2)(3,6)(4,8)(5,7)(9,13,15,12)(10,11,17,19)(14,20,24,21)(16,23,22,18)(25,26)(27,31)(28,39)(29,40)(30,34)(32,37)(33,36)(35,38);(1,4,5,6)(2,3,7,8)(9,11)(10,13)(12,17)(14,23)(15,19)(16,20)(18,24)(21,22)(25,28)(26,32)(27,36)(29,38)(30,37)(31,40)(33,35)(34,39);(1,3)(2,4)(5,8)(6,7)(9,14)(10,16)(11,18)(12,20)(13,21)(15,24)(17,22)(19,23)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,39,35,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9136644);