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 | |
4422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,19)(11,12,22,16)(13,17,21,24)(14,18,20,23)(25,26,31,33)(27,29,35,32)(28,30,39,34)(36,38,40,37);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,27,31,35)(26,32,33,29)(28,36,39,40)(30,37,34,38);(1,2,4,6)(3,7,8,5)(13,21)(14,20)(17,24)(18,23)(26,33)(29,32)(30,34)(37,38);(3,8)(5,7)(9,12,15,16)(10,11,19,22)(13,18,21,23)(14,24,20,17)(25,28)(26,34)(27,36)(29,37)(30,33)(31,39)(32,38)(35,40);(1,3)(2,5)(4,8)(6,7)(9,13,15,21)(10,17,19,24)(11,20,22,14)(12,23,16,18)(25,29)(26,35)(27,33)(28,38)(30,40)(31,32)(34,36)(37,39);(1,2)(3,5)(4,6)(7,8)(9,14,15,20)(10,18,19,23)(11,21,22,13)(12,24,16,17)(25,30,31,34)(26,28,33,39)(27,37,35,38)(29,40,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8850528);