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 | |
4612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,26,31,33)(27,30,38,32)(28,29,35,34)(36,37,40,39);(2,5)(3,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,24)(21,22)(25,27,31,38)(26,32,33,30)(28,36,35,40)(29,39,34,37);(1,3,6,8)(2,7,5,4)(10,14)(11,13)(17,20)(19,22)(26,33)(29,34)(30,32)(37,39);(1,4)(2,3)(5,8)(6,7)(9,12,15,21)(10,17,14,20)(11,19,13,22)(16,23,18,24)(25,28)(26,34)(27,36)(29,33)(30,39)(31,35)(32,37)(38,40);(1,5,6,2)(3,4,8,7)(9,13,15,11)(10,18,14,16)(12,19,21,22)(17,23,20,24)(25,29,31,34)(26,35,33,28)(27,37,38,39)(30,40,32,36);(1,4)(2,3)(5,8)(6,7)(9,14,15,10)(11,18,13,16)(12,20,21,17)(19,24,22,23)(25,30)(26,27)(28,39)(29,40)(31,32)(33,38)(34,36)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=ghi,d^2=gh,e^2=h,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, 8853442);