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 | |
4519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,31,34)(27,29,33,39)(28,30,38,32)(35,40,36,37);(1,2,5,4)(3,6,8,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27,31,33)(26,29,34,39)(28,36,38,35)(30,37,32,40);(1,2,5,4)(3,7,8,6)(10,15)(11,18)(16,21)(19,24)(25,28,31,38)(26,32,34,30)(27,35,33,36)(29,37,39,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,28)(26,30)(27,36)(29,37)(31,38)(32,34)(33,35)(39,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29,31,39)(26,33,34,27)(28,37,38,40)(30,35,32,36);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,30)(26,28)(27,37)(29,36)(31,32)(33,40)(34,38)(35,39)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592727);