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 | |
4452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,17)(11,18,12,15)(13,16,22,21)(19,24,20,23);(3,7)(6,8)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,26,31,35)(27,36,38,32)(28,30,34,40)(29,33,39,37);(1,2,5,4)(3,6,7,8)(25,27,31,38)(26,32,35,36)(28,29,34,39)(30,37,40,33);(9,10)(11,15)(12,18)(13,21)(14,17)(16,22)(19,24)(20,23)(25,28)(26,30)(27,29)(31,34)(32,37)(33,36)(35,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,15,17,18)(13,19,22,20)(16,24,21,23)(25,29)(26,33)(27,34)(28,38)(30,32)(31,39)(35,37)(36,40);(1,4,5,2)(3,8,7,6)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,30)(26,34)(27,37)(28,35)(29,36)(31,40)(32,39)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9129332);