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 | |
4356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(25,26)(27,33)(28,30)(29,39)(31,34)(32,38)(35,40)(36,37);(2,6)(3,7)(9,10,14,15)(11,13,17,22)(12,16,19,23)(18,21,20,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,34)(39,40);(1,2,5,6)(3,8,7,4)(10,15)(12,19)(13,22)(18,20)(25,28,31,29)(26,30,34,39)(27,35,38,36)(32,37,33,40);(2,6)(3,7)(9,11,14,17)(10,13,15,22)(12,20,19,18)(16,24,23,21)(25,29,31,28)(26,30,34,39)(27,36,38,35)(32,37,33,40);(1,3,5,7)(2,4,6,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,27)(26,33)(28,36)(29,35)(30,37)(31,38)(32,34)(39,40);(1,4)(2,7)(3,6)(5,8)(9,13)(10,17)(11,15)(12,21)(14,22)(16,20)(18,23)(19,24)(25,30)(26,28)(27,37)(29,34)(31,39)(32,35)(33,36)(38,40)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9415778);