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 | |
4276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,12)(11,16,13,17)(14,18,22,20)(19,23,21,24)(25,26,31,34)(27,35,38,32)(28,39,36,30)(29,37,40,33);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27,31,38)(26,32,34,35)(28,29,36,40)(30,37,39,33);(2,5)(6,8)(9,11,15,13)(10,17,12,16)(14,19,22,21)(18,24,20,23)(25,28,31,36)(26,30,34,39)(27,29,38,40)(32,37,35,33);(3,7)(6,8)(9,12,15,10)(11,16,13,17)(14,20,22,18)(19,23,21,24)(25,27)(26,32)(28,29)(30,37)(31,38)(33,39)(34,35)(36,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,11)(10,16,12,17)(14,21,22,19)(18,23,20,24)(25,29)(26,33)(27,36)(28,38)(30,32)(31,40)(34,37)(35,39);(1,2)(3,6)(4,5)(7,8)(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30)(26,28)(27,37)(29,32)(31,39)(33,38)(34,36)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 8183542);