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 | |
4445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,5,7)(27,33)(30,34)(35,39)(38,40);(1,3,6,5)(2,7,8,4)(9,10)(11,16)(12,13)(14,15)(17,21)(18,22)(19,24)(20,23)(25,26,29,32)(27,34,33,30)(28,31,37,36)(35,40,39,38);(2,8)(4,7)(10,15)(12,21)(16,23)(18,24)(25,27,29,33)(26,30,32,34)(28,35,37,39)(31,38,36,40);(2,8)(3,5)(9,11)(10,16)(12,18)(13,22)(14,20)(15,23)(17,19)(21,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,34)(35,40)(38,39);(1,4,6,7)(2,3,8,5)(9,12)(10,17)(11,18)(13,15)(14,21)(16,19)(20,24)(22,23)(25,27)(26,30)(28,35)(29,33)(31,38)(32,34)(36,40)(37,39);(1,5,6,3)(2,7,8,4)(9,13,14,17)(10,12,15,21)(11,19,20,22)(16,24,23,18)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,e^2=g,f^2=hi,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, 8540893);