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 | |
4545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10)(11,16)(12,17)(13,15)(14,18)(19,23)(20,24)(21,22)(25,26,31,34)(27,32,35,29)(28,30,33,39)(36,37,40,38);(1,3)(2,7)(4,5)(6,8)(9,11,15,19)(10,16,13,23)(12,20,22,18)(14,17,24,21)(25,27)(26,32)(28,36)(29,34)(30,37)(31,35)(33,40)(38,39);(1,4,6,7)(2,3,5,8)(10,13)(11,19)(17,21)(18,20)(27,35)(29,32)(36,40)(37,38);(2,5)(3,8)(9,12,15,22)(10,17,13,21)(11,18,19,20)(14,23,24,16)(25,28)(26,30)(27,36)(29,38)(31,33)(32,37)(34,39)(35,40);(1,4)(2,3)(5,8)(6,7)(9,13,15,10)(11,16,19,23)(12,21,22,17)(14,20,24,18)(25,29)(26,27)(28,38)(30,36)(31,32)(33,37)(34,35)(39,40);(1,5,6,2)(3,4,8,7)(9,14)(10,18)(11,17)(12,16)(13,20)(15,24)(19,21)(22,23)(25,30)(26,33)(27,37)(28,34)(29,36)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8496080);