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 | |
4263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,5,8,7)(9,10)(11,18)(12,21)(13,16)(14,17)(15,23)(19,22)(20,24)(25,26)(27,32)(28,34)(29,33)(30,35)(31,36)(37,38)(39,40);(1,3)(2,7)(4,5)(6,8)(9,11)(10,13)(12,19)(14,16)(15,22)(17,18)(20,23)(21,24)(25,27,31,39)(26,32,36,40)(28,29,35,37)(30,38,34,33);(1,2,6,4)(3,7,8,5)(9,12)(10,15)(11,19)(13,22)(14,23)(16,20)(17,21)(18,24)(25,28)(26,30)(27,37)(29,39)(31,35)(32,33)(34,36)(38,40);(2,4)(3,8)(9,12)(10,15)(11,20)(13,24)(14,23)(16,19)(17,21)(18,22)(25,29,31,37)(26,33,36,38)(27,28,39,35)(30,32,34,40);(1,4,6,2)(3,7,8,5)(9,13,14,18)(10,16,17,11)(12,22,23,24)(15,20,21,19)(25,28)(26,34)(27,29)(30,36)(31,35)(32,33)(37,39)(38,40);(1,5,6,7)(2,3,4,8)(9,10)(11,13)(12,15)(14,17)(16,18)(19,22)(20,24)(21,23)(25,30,31,34)(26,35,36,28)(27,38,39,33)(29,32,37,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227578);