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 | |
4216 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,13)(14,24)(15,17)(16,21)(18,23)(19,22);(1,3,5,8)(2,6,4,7)(9,11,15,21)(10,16,17,20)(12,22,23,14)(13,24,18,19)(25,26,30,31)(27,32,37,40)(28,33,38,36)(29,34,39,35);(1,3)(2,7)(4,6)(5,8)(10,17)(11,21)(12,23)(19,24)(26,31)(29,39)(32,40)(33,36);(1,3,5,8)(2,6,4,7)(9,12)(10,18)(11,22)(13,17)(14,21)(15,23)(16,19)(20,24)(25,27)(26,32)(28,34)(29,36)(30,37)(31,40)(33,39)(35,38);(1,3)(2,6)(4,7)(5,8)(9,13,15,18)(10,12,17,23)(11,19,21,24)(14,20,22,16)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(1,4,5,2)(3,7,8,6)(9,14,15,22)(10,19,17,24)(11,12,21,23)(13,20,18,16)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|b^2=i,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9348146);