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 | |
4408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,15,20)(11,21,22,16)(12,24,23,17)(13,18,19,14)(25,26,31,32)(27,33,37,28)(29,34,36,40)(30,35,39,38);(11,22)(12,23)(16,21)(17,24)(26,32)(28,33)(34,40)(35,38);(1,2,5,4)(3,6,7,8)(9,11,15,22)(10,16,20,21)(12,13,23,19)(14,17,18,24)(25,27,31,37)(26,33,32,28)(29,30,36,39)(34,35,40,38);(3,7)(6,8)(9,12)(10,17)(11,13)(14,21)(15,23)(16,18)(19,22)(20,24)(25,28)(26,27)(29,38)(30,34)(31,33)(32,37)(35,36)(39,40);(1,3,5,7)(2,6,4,8)(9,13)(10,18)(11,23)(12,22)(14,20)(15,19)(16,24)(17,21)(25,29)(26,34)(27,30)(28,38)(31,36)(32,40)(33,35)(37,39);(1,4,5,2)(3,8,7,6)(9,14,15,18)(10,19,20,13)(11,17,22,24)(12,21,23,16)(25,30)(26,35)(27,36)(28,34)(29,37)(31,39)(32,38)(33,40)Polycyclic presentation:
2^9|a^2=gi,c^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10413487);