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 |
| prime radical | #58924 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,15,12)(11,18,14,16)(13,17,21,20)(19,24,22,23)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,40)(38,39);(1,3,5,8)(2,6,7,4)(13,21)(17,20)(19,22)(23,24)(25,27,31,33)(26,29,35,36)(28,37,38,30)(32,34,39,40);(1,3)(2,4)(5,8)(6,7)(9,11,15,14)(10,16,12,18)(13,19,21,22)(17,23,20,24)(25,28,31,38)(26,32,35,39)(27,37,33,30)(29,34,36,40);(1,4)(2,3)(5,6)(7,8)(9,12,15,10)(11,18,14,16)(13,20,21,17)(19,24,22,23)(25,26)(27,29)(28,32)(30,40)(31,35)(33,36)(34,37)(38,39);(9,13,15,21)(10,17,12,20)(11,19,14,22)(16,23,18,24)(25,29)(26,33)(27,35)(28,34)(30,32)(31,36)(37,39)(38,40);(9,14,15,11)(10,18,12,16)(13,22,21,19)(17,24,20,23)(25,30)(26,34)(27,28)(29,39)(31,37)(32,36)(33,38)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=ghi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9143665);