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 | #37820 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10,15,13)(11,14,17,19)(12,16,21,22)(18,20,23,24)(25,26,32,29)(27,33,30,39)(28,37,36,34)(31,35,38,40);(2,7)(3,4)(9,11)(10,14)(12,18)(13,19)(15,17)(16,20)(21,23)(22,24)(25,27,32,30)(26,33,29,39)(28,31,36,38)(34,40,37,35);(1,3,6,4)(2,8,7,5)(11,17)(14,19)(18,23)(20,24)(25,28)(26,34)(27,31)(29,37)(30,38)(32,36)(33,40)(35,39);(1,4,6,3)(2,5,7,8)(9,12)(10,16)(11,18)(13,22)(14,20)(15,21)(17,23)(19,24)(25,29,32,26)(27,33,30,39)(28,37,36,34)(31,40,38,35);(1,5)(2,3)(4,7)(6,8)(9,13,15,10)(11,19,17,14)(12,16,21,22)(18,20,23,24)(25,30,32,27)(26,33,29,39)(28,38,36,31)(34,40,37,35);(9,14,15,19)(10,17,13,11)(12,20,21,24)(16,23,22,18)(25,31,32,38)(26,35,29,40)(27,36,30,28)(33,34,39,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8248821);