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 | |
4487 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)(11,19)(12,15)(13,17)(14,18)(16,22)(20,24)(21,23)(25,26,31,33)(27,30,38,32)(28,29,35,34)(36,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,27,31,38)(26,32,33,30)(28,36,35,40)(29,39,34,37);(2,5)(3,7)(13,21)(14,20)(17,23)(18,24)(26,33)(29,34)(30,32)(37,39);(3,7)(6,8)(9,12,15,10)(11,19,22,16)(13,17,21,23)(14,24,20,18)(25,28)(26,34)(27,36)(29,33)(30,39)(31,35)(32,37)(38,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,18,19,24)(25,29,31,34)(26,35,33,28)(27,37,38,39)(30,40,32,36);(1,2)(3,6)(4,5)(7,8)(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,17,19,23)(25,30)(26,27)(28,39)(29,40)(31,32)(33,38)(34,36)(35,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=gh,e^2=g,f^2=gh,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, 8843887);