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 | |
4469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26,31,33)(27,32,35,29)(28,34,37,39)(30,36,40,38);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,17,18)(13,22,24,20)(14,21,19,23)(25,27)(26,32)(28,30)(29,33)(31,35)(34,36)(37,40)(38,39);(4,8)(6,7)(10,17)(14,19)(16,18)(21,23)(26,33)(27,35)(30,40)(34,39);(2,3)(6,7)(9,12,15,11)(10,18,17,16)(13,20,24,22)(14,23,19,21)(25,28)(26,34)(27,30)(29,38)(31,37)(32,36)(33,39)(35,40);(1,3,5,2)(4,7,8,6)(9,13)(10,19)(11,20)(12,22)(14,17)(15,24)(16,21)(18,23)(25,29,31,32)(26,35,33,27)(28,36,37,38)(30,39,40,34);(1,4,5,8)(2,6,3,7)(9,14,15,19)(10,13,17,24)(11,21,12,23)(16,22,18,20)(25,30)(26,36)(27,28)(29,39)(31,40)(32,34)(33,38)(35,37)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8300959);