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 | |
4459 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)(3,5)(4,7)(6,8)(9,10)(11,12)(13,16)(14,22)(15,17)(18,23)(19,24)(20,21)(27,33)(28,36)(31,34)(32,37);(1,3)(2,4)(5,8)(6,7)(9,11,15,13)(10,16,17,12)(14,19,23,21)(18,24,22,20)(25,26)(27,34)(28,37)(29,30)(31,33)(32,36)(35,39)(38,40);(1,4,6,5)(2,7,8,3)(10,17)(12,16)(14,23)(19,21)(26,30)(27,33)(32,37)(35,38);(3,7)(4,5)(9,12,15,16)(10,11,17,13)(14,20,23,24)(18,19,22,21)(25,27)(26,31)(28,38)(29,33)(30,34)(32,40)(35,36)(37,39);(1,5,6,4)(2,7,8,3)(9,13,15,11)(10,12,17,16)(14,21,23,19)(18,20,22,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,39)(33,38)(34,40);(1,2)(3,4)(5,7)(6,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,24)(17,22)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,37)(38,40)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840391);