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 | |
4506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(11,17)(15,18)(19,23)(22,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,40)(38,39);(3,7)(5,8)(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(25,27,31,40)(26,32,36,37)(28,38,35,34)(29,30,39,33);(1,2)(3,5)(4,6)(7,8)(10,14)(11,17)(12,20)(22,24)(28,35)(29,39)(30,33)(34,38);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,31,35)(26,33,36,30)(27,38,40,34)(29,37,39,32);(1,3,4,7)(2,5,6,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,29,31,39)(26,34,36,38)(27,33,40,30)(28,32,35,37);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,30,31,33)(26,35,36,28)(27,39,40,29)(32,34,37,38)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8964610);