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 | |
4482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26)(27,36)(28,34)(29,30)(31,33)(32,35)(37,39)(38,40);(1,2)(3,8)(4,6)(5,7)(12,19)(15,17)(20,24)(22,23)(25,27,32,31)(26,33,35,36)(28,37,29,38)(30,39,34,40);(1,3,5,4)(2,6,7,8)(10,11)(15,17)(16,18)(22,23)(25,28,32,29)(26,34,35,30)(27,37,31,38)(33,40,36,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,29,32,28)(26,34,35,30)(27,38,31,37)(33,40,36,39);(1,4,5,3)(2,6,7,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,30,32,34)(26,29,35,28)(27,39,31,40)(33,38,36,37);(1,2)(3,6)(4,8)(5,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,31,32,27)(26,33,35,36)(28,38,29,37)(30,39,34,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gi,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10065301);