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 | |
4393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,18,14,16)(12,21,22,13)(19,24,23,20)(25,26)(27,35)(28,30)(29,32)(31,34)(33,40)(36,37)(38,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,17,18)(12,19,22,23)(13,24,21,20)(27,29)(32,35)(36,38)(37,39);(1,3,5,4)(2,6,7,8)(10,17)(12,22)(16,18)(19,23)(25,27,31,29)(26,32,34,35)(28,38,33,36)(30,37,40,39);(3,4)(6,8)(9,12,15,22)(10,13,17,21)(11,19,14,23)(16,24,18,20)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,39,35,37);(1,4,5,3)(2,6,7,8)(9,13)(10,12)(11,20)(14,24)(15,21)(16,23)(17,22)(18,19)(25,29,31,27)(26,32,34,35)(28,36,33,38)(30,37,40,39);(1,2)(3,6)(4,8)(5,7)(9,14,15,11)(10,18,17,16)(12,23,22,19)(13,20,21,24)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,36)(35,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9071314);