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 | |
4543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(25,26)(27,30)(28,34)(29,39)(31,33)(32,35)(36,37)(38,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,27,32,31)(26,33,35,30)(28,36,29,38)(34,40,39,37);(1,3)(2,5)(4,8)(6,7)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,28,32,29)(26,34,35,39)(27,36,31,38)(30,37,33,40);(2,6)(5,7)(9,12,14,11)(10,16,18,15)(13,19,22,20)(17,23,21,24)(25,29,32,28)(26,34,35,39)(27,36,31,38)(30,40,33,37);(1,3)(2,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,30)(26,27)(28,37)(29,40)(31,35)(32,33)(34,36)(38,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,31,32,27)(26,33,35,30)(28,38,29,36)(34,40,39,37)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8597147);