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 | |
4530 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)(3,7)(4,6)(5,8)(9,10,14,15)(11,17,19,12)(13,21,22,16)(18,20,24,23)(25,26,31,35)(27,36,37,32)(28,38,40,33)(29,30,34,39);(1,2,4,6)(3,5,8,7)(9,11,14,19)(10,12,15,17)(13,18,22,24)(16,20,21,23)(25,27,31,37)(26,32,35,36)(28,39,40,30)(29,38,34,33);(1,2)(3,5)(4,6)(7,8)(10,15)(12,17)(13,22)(18,24)(28,40)(29,34)(30,39)(33,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,19)(13,16)(14,15)(18,23)(20,24)(21,22)(25,28)(26,33)(27,30)(29,36)(31,40)(32,34)(35,38)(37,39);(9,12,14,17)(10,11,15,19)(13,20,22,23)(16,18,21,24)(25,29,31,34)(26,30,35,39)(27,33,37,38)(28,32,40,36);(9,13,14,22)(10,16,15,21)(11,18,19,24)(12,20,17,23)(25,30)(26,34)(27,28)(29,35)(31,39)(32,33)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8970472);