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 | |
4442 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)(11,17)(12,16)(13,21)(14,15)(18,22)(19,23)(20,24)(25,26,31,34)(27,32,36,30)(28,37,29,33)(35,40,38,39);(1,2)(3,5)(4,6)(7,8)(10,15)(16,17)(18,21)(23,24)(25,27)(26,32)(28,38)(29,35)(30,34)(31,36)(33,40)(37,39);(1,2,4,6)(3,7,8,5)(9,11,14,12)(10,16,15,17)(13,20,22,19)(18,24,21,23)(25,28,31,29)(26,33,34,37)(27,35,36,38)(30,40,32,39);(3,8)(5,7)(9,12,14,11)(10,17,15,16)(13,20,22,19)(18,24,21,23)(25,29,31,28)(26,33,34,37)(27,35,36,38)(30,39,32,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,23)(21,22)(25,30)(26,27)(28,39)(29,40)(31,32)(33,38)(34,36)(35,37);(1,2)(3,5)(4,6)(7,8)(9,13,14,22)(10,18,15,21)(11,19,12,20)(16,23,17,24)(25,27)(26,30)(28,38)(29,35)(31,36)(32,34)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8850162);