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 | |
4430 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)(9,10,14,17)(11,18,12,15)(13,16,22,21)(19,24,20,23)(25,26,31,37)(27,32,39,40)(28,34,38,33)(29,35,36,30);(1,2,4,6)(3,5,8,7)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,27,31,39)(26,32,37,40)(28,36,38,29)(30,33,35,34);(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,15,12,18)(13,21,22,16)(19,24,20,23)(25,28,31,38)(26,33,37,34)(27,29,39,36)(30,40,35,32);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,24)(21,22)(25,28)(26,34)(27,36)(29,39)(30,32)(31,38)(33,37)(35,40);(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,29,31,36)(26,35,37,30)(27,38,39,28)(32,33,40,34);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,30,31,35)(26,36,37,29)(27,33,39,34)(28,32,38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8922487);