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 | #39440 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10,14,15)(11,17,18,16)(12,13,21,23)(19,20,22,24);(1,3,6,5)(2,7,4,8)(10,15)(12,21)(16,17)(19,22)(25,26,30,31)(27,32,34,39)(28,29,33,38)(35,40,37,36);(1,3,6,5)(2,8,4,7)(10,15)(11,18)(13,23)(19,22)(26,31)(27,34)(29,38)(35,37);(9,11)(10,16)(12,22)(13,20)(14,18)(15,17)(19,21)(23,24)(25,27)(26,32)(28,37)(29,36)(30,34)(31,39)(33,35)(38,40);(1,4,6,2)(3,7,5,8)(9,12)(10,13)(11,19)(14,21)(15,23)(16,24)(17,20)(18,22)(25,28,30,33)(26,29,31,38)(27,35,34,37)(32,40,39,36);(1,5,6,3)(2,7,4,8)(9,13)(10,12)(11,20)(14,23)(15,21)(16,22)(17,19)(18,24)(25,29)(26,33)(27,36)(28,31)(30,38)(32,35)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8826673);