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 | |
4611 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)(11,16)(12,14)(13,20)(15,19)(17,21)(18,22)(23,24)(26,31)(27,32)(33,38)(34,39);(1,3,5,8)(2,6,4,7)(9,11,14,19)(10,15,12,16)(13,18,21,24)(17,22,20,23)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(1,3)(2,7)(4,6)(5,8)(10,12)(13,21)(15,16)(18,24)(29,37)(33,38)(34,39)(35,40);(1,3,5,8)(2,6,4,7)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40)Polycyclic presentation:
2^9|b^2=gi,d^2=h,f^2=gh,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, 9348156);