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 | |
4451 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)(11,16)(12,19)(13,21)(14,15)(17,23)(18,20)(22,24)(25,26,31,28)(27,29,32,35)(30,36,39,33)(34,38,40,37);(1,2)(3,8)(4,6)(5,7)(9,11,14,19)(10,12,15,16)(13,18,23,22)(17,20,21,24)(25,27)(26,29)(28,35)(30,34)(31,32)(33,37)(36,38)(39,40);(1,3,5,6)(2,4,7,8)(10,15)(12,16)(17,21)(20,24)(26,28)(27,32)(33,36)(34,40);(1,2,5,7)(3,4,6,8)(9,12)(10,11)(13,20)(14,16)(15,19)(17,18)(21,22)(23,24)(25,28,31,26)(27,29,32,35)(30,36,39,33)(34,37,40,38);(1,2)(3,4)(5,7)(6,8)(9,11)(10,16)(12,15)(13,22)(14,19)(17,20)(18,23)(21,24)(25,29)(26,32)(27,28)(30,38)(31,35)(33,34)(36,40)(37,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,18)(12,20)(14,23)(15,21)(16,24)(19,22)(25,30)(26,33)(27,34)(28,36)(29,37)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=ghi,d^2=i,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, 10157486);