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 | #37154 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(12,20)(15,21)(18,24)(22,23)(25,26)(27,34)(28,31)(29,37)(30,33)(32,39)(35,36)(38,40);(9,10)(11,16)(12,21)(13,17)(14,19)(15,20)(18,22)(23,24)(25,26,30,33)(27,28,31,34)(29,32,39,37)(35,36,40,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,27,30,31)(26,28,33,34)(29,35,39,40)(32,36,37,38);(3,7)(5,8)(9,12,13,20)(10,15,17,21)(11,18,19,24)(14,22,16,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,28)(26,31)(27,33)(29,38)(30,34)(32,35)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,29)(26,32)(27,35)(28,36)(30,39)(31,40)(33,37)(34,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10387938);