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 | |
4499 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)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23)(25,26,32,36)(27,29,30,33)(28,35,31,34)(37,38,40,39);(1,2)(3,7)(4,5)(6,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27)(26,33)(28,37)(29,36)(30,32)(31,40)(34,38)(35,39);(2,5)(6,7)(9,10,13,16)(11,17,19,14)(12,15,22,20)(18,24,21,23)(25,28,32,31)(26,34,36,35)(27,37,30,40)(29,39,33,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,29,32,33)(26,27,36,30)(28,38,31,39)(34,40,35,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,30,32,27)(26,33,36,29)(28,37,31,40)(34,39,35,38);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,31,32,28)(26,35,36,34)(27,37,30,40)(29,39,33,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8665284);