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 | |
4887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,26)(27,32)(28,34)(29,37)(30,35)(31,36)(33,38)(39,40);(1,3,4,7)(2,5,6,8)(10,11)(15,17)(16,18)(23,24)(25,27,31,39)(26,32,36,40)(28,30,38,37)(29,34,35,33);(1,2)(3,5)(4,6)(7,8)(25,28)(26,33)(27,37)(29,40)(30,39)(31,38)(32,35)(34,36);(1,3)(2,5)(4,7)(6,8)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,28)(26,34)(27,30)(29,40)(31,38)(32,35)(33,36)(37,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,29)(26,30)(27,34)(28,40)(31,35)(32,38)(33,39)(36,37);(9,13)(10,16)(11,18)(12,19)(14,22)(15,23)(17,24)(20,21)(25,30)(26,35)(27,38)(28,39)(29,36)(31,37)(32,33)(34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=g,e^2=g,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, 8598157);