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 | #58887 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,16,20)(11,21,24,17)(12,18,13,15)(14,19,23,22)(25,26,30,33)(27,31,37,34)(28,29,32,38)(35,36,39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,17)(12,19)(13,22)(14,18)(15,23)(16,24)(20,21)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3)(2,7)(4,6)(5,8)(9,12,16,13)(10,18,20,15)(11,19,24,22)(14,21,23,17)(25,26,30,33)(27,34,37,31)(28,29,32,38)(35,40,39,36);(1,3,5,8)(2,6,4,7)(9,13,16,12)(10,18,20,15)(11,22,24,19)(14,21,23,17)(25,28)(26,29)(27,35)(30,32)(31,36)(33,38)(34,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,14,16,23)(10,19,20,22)(11,18,24,15)(12,21,13,17)(25,29)(26,32)(27,36)(28,33)(30,38)(31,39)(34,35)(37,40);(1,4,5,2)(3,7,8,6)(9,15,16,18)(10,12,20,13)(11,23,24,14)(17,19,21,22)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9352450);