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 | |
4378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,15,13)(11,16,22,21)(12,18,23,17)(14,24,20,19)(25,26,30,33)(27,29,32,35)(28,31,38,40)(34,37,36,39);(1,3,6,5)(2,7,4,8)(9,11,15,22)(10,16,13,21)(12,20,23,14)(17,24,18,19)(25,27)(26,29)(28,36)(30,32)(31,39)(33,35)(34,38)(37,40);(9,12)(10,17)(11,20)(13,18)(14,22)(15,23)(16,24)(19,21)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,40)(35,39);(1,2,6,4)(3,8,5,7)(9,12)(10,18)(11,20)(13,17)(14,22)(15,23)(16,19)(21,24)(25,26)(27,35)(28,31)(29,32)(30,33)(34,39)(36,37)(38,40);(1,4,6,2)(3,8,5,7)(9,13,15,10)(11,21,22,16)(12,18,23,17)(14,24,20,19)(25,29,30,35)(26,32,33,27)(28,37,38,39)(31,36,40,34);(1,5,6,3)(2,7,4,8)(9,14,15,20)(10,19,13,24)(11,12,22,23)(16,17,21,18)(25,28)(26,31)(27,36)(29,39)(30,38)(32,34)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10413480);