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 | #49704 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,18)(12,15)(13,20)(14,16)(17,22)(19,23)(21,24)(25,26,32,30)(27,34,31,29)(28,33,38,40)(35,36,37,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27,32,31)(26,29,30,34)(28,35,38,37)(33,39,40,36);(1,2,4,6)(3,5,7,8)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,28)(26,33)(27,35)(29,39)(30,40)(31,37)(32,38)(34,36);(1,2)(3,5)(4,6)(7,8)(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24)(25,29)(26,31)(27,30)(28,36)(32,34)(33,35)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,32,26)(27,34,31,29)(28,33,38,40)(35,39,37,36);(1,3)(2,5)(4,7)(6,8)(9,14,15,11)(10,16,12,18)(13,21,22,19)(17,23,20,24)(25,31,32,27)(26,34,30,29)(28,37,38,35)(33,36,40,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034187);