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 | |
4381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,23,21)(25,26,31,34)(27,35,38,32)(28,30,36,40)(29,33,39,37);(1,2,4,6)(3,5,8,7)(9,11,14,19)(10,15,17,12)(13,18,22,23)(16,21,24,20);(1,2)(3,5)(4,6)(7,8)(25,27,31,38)(26,32,34,35)(28,29,36,39)(30,37,40,33);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,21)(14,15)(16,23)(17,19)(18,24)(20,22)(25,28)(26,30)(27,29)(31,36)(32,37)(33,35)(34,40)(38,39);(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23)(25,29)(26,33)(27,36)(28,38)(30,32)(31,39)(34,37)(35,40);(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,24)(19,23)(25,30)(26,28)(27,37)(29,32)(31,40)(33,38)(34,36)(35,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319359);