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 | |
4287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,20)(12,22)(13,14)(15,17)(16,21)(18,24)(19,23)(25,26,31,34)(27,29,33,30)(28,36,39,32)(35,38,40,37);(1,2)(3,6)(4,5)(7,8)(9,11,15,21)(10,16,17,20)(12,23,24,13)(14,18,19,22)(25,27,31,33)(26,30,34,29)(28,35,39,40)(32,38,36,37);(2,5)(6,8)(10,17)(14,19)(16,20)(18,22)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(3,7)(6,8)(9,12,15,24)(10,18,17,22)(11,13,21,23)(14,20,19,16)(25,28,31,39)(26,32,34,36)(27,35,33,40)(29,37,30,38);(1,3)(2,6)(4,7)(5,8)(9,13)(10,19)(11,12)(14,17)(15,23)(16,22)(18,20)(21,24)(25,29,31,30)(26,33,34,27)(28,37,39,38)(32,40,36,35);(1,2)(3,6)(4,5)(7,8)(9,14,15,19)(10,13,17,23)(11,18,21,22)(12,20,24,16)(25,30,31,29)(26,27,34,33)(28,38,39,37)(32,35,36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8180644);