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 | |
4446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,15)(11,16,19,14)(12,20,22,17)(18,23,21,24)(25,26)(27,32)(28,34)(29,35)(30,38)(31,33)(36,39)(37,40);(1,2)(3,4)(5,7)(6,8)(9,11,13,19)(10,14,15,16)(12,18,22,21)(17,23,20,24)(25,27,31,36)(26,32,33,39)(28,37,38,29)(30,35,34,40);(1,3,5,8)(2,6,7,4)(10,15)(14,16)(17,20)(23,24)(26,33)(27,36)(28,38)(35,40);(3,8)(4,6)(9,10)(11,14)(12,17)(13,15)(16,19)(18,23)(20,22)(21,24)(25,28)(26,34)(27,37)(29,36)(30,33)(31,38)(32,40)(35,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,21)(13,19)(14,15)(17,23)(18,22)(20,24)(25,29,31,37)(26,35,33,40)(27,38,36,28)(30,39,34,32);(1,2)(3,6)(4,8)(5,7)(9,12)(10,17)(11,18)(13,22)(14,23)(15,20)(16,24)(19,21)(25,30,31,34)(26,28,33,38)(27,35,36,40)(29,32,37,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9046652);