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 | |
4583 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)(3,5)(4,6)(7,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,31,33)(27,32,35,39)(28,30,37,40)(29,34,36,38);(1,3,7,6)(2,4,8,5)(12,19)(15,17)(20,24)(22,23)(25,27,31,35)(26,32,33,39)(28,36,37,29)(30,38,40,34);(1,4,7,5)(2,3,8,6)(10,11)(15,17)(16,18)(22,23)(26,33)(27,35)(28,37)(34,38);(2,8)(4,5)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28)(26,30)(27,36)(29,35)(31,37)(32,38)(33,40)(34,39);(1,5,7,4)(2,3,8,6)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,31,36)(26,34,33,38)(27,37,35,28)(30,32,40,39);(1,6,7,3)(2,5,8,4)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,28)(27,38)(29,39)(31,40)(32,36)(33,37)(34,35)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=g,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10078918);