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 |
| prime radical | #58300 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4346 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)(9,10,15,12)(11,16,13,17)(14,18,22,20)(19,23,21,24)(25,26,31,34)(27,32,37,28)(29,30,33,39)(35,36,40,38);(2,5)(6,7)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27,31,37)(26,32,34,28)(29,35,33,40)(30,36,39,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,17,12,16)(14,19,22,21)(18,24,20,23)(25,28,31,32)(26,27,34,37)(29,38,33,36)(30,35,39,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,10)(11,16,13,17)(14,20,22,18)(19,23,21,24)(25,29)(26,30)(27,35)(28,38)(31,33)(32,36)(34,39)(37,40);(1,3,4,8)(2,6,5,7)(9,13,15,11)(10,16,12,17)(14,21,22,19)(18,23,20,24)(25,28)(26,27)(29,38)(30,35)(31,32)(33,36)(34,37)(39,40);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30)(26,33)(27,36)(28,35)(29,34)(31,39)(32,40)(37,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440852);