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 | |
4480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,12)(13,21)(14,15)(16,20)(17,22)(18,24)(19,23)(25,26,30,34)(27,35,37,31)(28,32,38,36)(29,39,40,33);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,12,15,20)(13,18,22,23)(17,19,21,24)(25,27)(26,31)(28,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(17,21)(18,23)(25,26,30,34)(27,31,37,35)(28,32,38,36)(29,33,40,39);(2,5)(6,7)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,24)(22,23)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,21)(22,24)(25,28)(26,32)(27,33)(29,35)(30,38)(31,40)(34,36)(37,39);(1,3)(2,6)(4,8)(5,7)(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,19,20,24)(25,29,30,40)(26,33,34,39)(27,36,37,32)(28,35,38,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=ghi,f^2=gh,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^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066086);