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 | |
4243 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)(3,4)(5,8)(6,7)(9,10,15,13)(11,19,21,16)(12,22,23,17)(14,24,20,18)(25,26,31,34)(27,29,33,30)(28,36,39,32)(35,38,40,37);(1,3)(2,8)(4,6)(5,7)(9,11,15,21)(10,16,13,19)(12,20,23,14)(17,18,22,24)(25,27,31,33)(26,30,34,29)(28,35,39,40)(32,38,36,37);(1,2,7,6)(3,8,5,4)(9,12,15,23)(10,17,13,22)(11,14,21,20)(16,24,19,18)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(1,4)(2,5)(3,6)(7,8)(9,12)(10,17)(11,20)(13,22)(14,21)(15,23)(16,18)(19,24)(25,28,31,39)(26,32,34,36)(27,35,33,40)(29,37,30,38);(1,5,7,3)(2,8,6,4)(9,13,15,10)(11,16,21,19)(12,22,23,17)(14,24,20,18)(25,29,31,30)(26,33,34,27)(28,37,39,38)(32,40,36,35);(1,6,7,2)(3,8,5,4)(9,14,15,20)(10,18,13,24)(11,12,21,23)(16,22,19,17)(25,30,31,29)(26,27,34,33)(28,38,39,37)(32,35,36,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150706);