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 | #44438 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,19)(11,20,21,16)(12,22,24,17)(13,14,18,23)(26,31)(27,36)(29,34)(37,39);(1,2)(3,7)(4,5)(6,8)(9,11,15,21)(10,16,19,20)(12,23,24,14)(13,22,18,17)(25,26)(27,32)(28,34)(29,35)(30,31)(33,36)(37,40)(38,39);(2,5)(3,8)(12,24)(13,18)(14,23)(17,22)(25,27,30,36)(26,32,31,33)(28,39,35,37)(29,40,34,38);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,14)(13,20)(15,24)(16,18)(19,22)(21,23)(25,27)(26,33)(28,39)(29,38)(30,36)(31,32)(34,40)(35,37);(1,3)(2,6)(4,8)(5,7)(9,13,15,18)(10,14,19,23)(11,17,21,22)(12,16,24,20)(25,28,30,35)(26,34,31,29)(27,37,36,39)(32,40,33,38);(1,2)(3,6)(4,5)(7,8)(9,14)(10,18)(11,12)(13,19)(15,23)(16,17)(20,22)(21,24)(25,29,30,34)(26,35,31,28)(27,38,36,40)(32,39,33,37)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8969798);