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 |
| prime radical | #50856 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4620 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)(11,12)(13,21)(14,18)(15,16)(17,24)(19,23)(20,22)(25,26)(27,28)(29,30)(31,33)(32,34)(35,36)(37,38)(39,40);(1,2,5,6)(3,7,8,4)(9,11,15,14)(10,12,16,18)(13,22,24,19)(17,23,21,20)(25,27)(26,29)(28,34)(30,32)(31,35)(33,37)(36,40)(38,39);(2,6)(4,7)(10,16)(11,14)(17,21)(19,22)(25,28,32,29)(26,27,34,30)(31,37,39,36)(33,38,40,35);(2,6)(3,8)(9,12,15,18)(10,11,16,14)(13,23,24,20)(17,22,21,19)(25,29,32,28)(26,27,34,30)(31,37,39,36)(33,35,40,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,19)(12,20)(14,22)(15,24)(16,21)(18,23)(25,30,32,27)(26,29,34,28)(31,38,39,35)(33,37,40,36);(1,4)(2,3)(5,7)(6,8)(9,14,15,11)(10,18,16,12)(13,22,24,19)(17,23,21,20)(25,31)(26,33)(27,35)(28,36)(29,37)(30,38)(32,39)(34,40)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8296171);