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 | |
4406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,17,22,24)(13,18,23,21)(25,26)(27,30)(28,29)(31,32)(33,40)(34,35)(36,38)(37,39);(1,2)(3,8)(4,6)(5,7)(10,15)(11,20)(17,24)(18,21)(25,27,31,33)(26,30,32,40)(28,36,37,34)(29,38,39,35);(1,3,5,6)(2,4,7,8)(26,32)(27,33)(28,37)(35,38);(1,2,5,7)(3,4,6,8)(9,11,14,20)(10,16,15,19)(12,21,22,18)(13,17,23,24)(25,28,31,37)(26,29,32,39)(27,34,33,36)(30,35,40,38);(1,2)(3,4)(5,7)(6,8)(9,12,14,22)(10,17,15,24)(11,18,20,21)(13,19,23,16)(25,29)(26,28)(27,35)(30,34)(31,39)(32,37)(33,38)(36,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,18)(11,17)(12,16)(14,23)(15,21)(19,22)(20,24)(25,30)(26,33)(27,32)(28,35)(29,36)(31,40)(34,39)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10157518);