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 | |
4289 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)(11,20)(12,13)(14,22)(15,16)(17,21)(18,24)(19,23)(25,26)(27,32)(28,37)(29,30)(31,34)(33,40)(35,39)(36,38);(1,2)(3,7)(4,5)(6,8)(10,16)(11,21)(13,18)(19,22)(25,27,31,28)(26,32,34,37)(29,38,33,35)(30,36,40,39);(2,5)(6,7)(9,11)(10,17)(12,22)(13,23)(14,18)(15,21)(16,20)(19,24)(25,28,31,27)(26,32,34,37)(29,38,33,35)(30,39,40,36);(1,2)(3,6)(4,5)(7,8)(9,12)(10,18)(11,22)(13,16)(14,17)(15,24)(19,21)(20,23);(1,2)(3,6)(4,5)(7,8)(9,13,15,18)(10,12,16,24)(11,23,21,14)(17,22,20,19)(25,29,31,33)(26,30,34,40)(27,35,28,38)(32,39,37,36);(1,3)(2,6)(4,8)(5,7)(9,14,15,23)(10,19,16,22)(11,18,21,13)(12,17,24,20)(25,30)(26,33)(27,36)(28,39)(29,34)(31,40)(32,35)(37,38)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8666276);