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 | |
4241 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)(3,7)(4,8)(5,6)(9,10,14,18)(11,19,21,15)(12,22,23,16)(13,17,20,24)(25,26)(27,29)(28,36)(30,34)(31,32)(33,39)(35,38)(37,40);(1,2,5,6)(3,4,8,7)(9,11,14,21)(10,15,18,19)(12,13,23,20)(16,17,22,24)(26,32)(27,34)(33,36)(35,40);(1,2,5,6)(3,7,8,4)(9,12,14,23)(10,16,18,22)(11,20,21,13)(15,24,19,17)(25,27,31,34)(26,29,32,30)(28,35,39,40)(33,37,36,38);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,13)(14,23)(15,17)(18,22)(19,24)(20,21)(25,28,31,39)(26,33,32,36)(27,35,34,40)(29,37,30,38);(1,2)(3,4)(5,6)(7,8)(9,13)(10,17)(11,12)(14,20)(15,16)(18,24)(19,22)(21,23)(25,29,31,30)(26,27,32,34)(28,37,39,38)(33,35,36,40);(1,4)(2,3)(5,7)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,24)(22,23)(25,30,31,29)(26,34,32,27)(28,38,39,37)(33,40,36,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9148183);