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 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,21)(13,17)(15,18)(16,20)(19,24)(22,23)(25,26)(27,34)(28,31)(29,37)(30,33)(32,39)(35,36)(38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,13)(15,19)(16,22)(17,21)(18,24)(20,23)(25,27,30,31)(26,28,33,34)(29,35,39,40)(32,36,37,38);(1,2)(3,5)(4,6)(7,8)(11,19)(13,23)(14,24)(17,22)(25,28,30,34)(26,27,33,31)(29,38,39,36)(32,40,37,35);(9,12)(10,16)(11,13)(14,22)(15,20)(17,24)(18,21)(19,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,13)(10,17)(11,20)(12,19)(14,16)(15,23)(18,22)(21,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,40)(35,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,14)(10,11)(12,22)(13,16)(15,24)(17,20)(18,19)(21,23)(25,29,30,39)(26,32,33,37)(27,35,31,40)(28,36,34,38)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8941655);