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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3997 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,13)(14,18)(15,19)(17,24)(20,23)(21,22)(25,26,30,32)(27,33,35,28)(29,37,38,31)(34,36,40,39);(1,2)(3,6)(4,5)(7,8)(11,19)(15,16)(20,22)(21,23)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(2,5)(3,7)(9,11,14,19)(10,15,18,16)(12,22,17,20)(13,23,24,21)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(3,7)(6,8)(9,12,14,17)(10,13,18,24)(11,20,19,22)(15,21,16,23)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,21)(14,19)(15,18)(17,20)(23,24);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,21)(12,18)(14,24)(15,22)(16,20)(19,23)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,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, 9422985);