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 | |
4655 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)(13,21)(16,22)(18,23)(19,24)(25,26,30,34)(27,35,37,31)(28,32,38,36)(29,39,40,33);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(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)(11,17)(16,22)(18,23)(25,26,30,34)(27,31,37,35)(28,32,38,36)(29,33,40,39);(2,5)(6,7)(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(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)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,28)(26,32)(27,33)(29,35)(30,38)(31,40)(34,36)(37,39);(1,3,4,8)(2,6,5,7)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(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,c^2=gh,d^2=i,f^2=g,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, 8066614);