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 | |
4224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10,16,15)(11,20,14,17)(12,18,13,19)(21,24,22,23)(25,26)(27,35)(28,32)(29,30)(31,34)(33,40)(36,37)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11,16,14)(10,17,15,20)(12,21,13,22)(18,23,19,24)(25,27,31,28)(26,32,34,35)(29,38,33,36)(30,37,40,39);(1,2)(3,4)(5,8)(6,7)(9,12,16,13)(10,18,15,19)(11,21,14,22)(17,23,20,24)(25,26,31,34)(27,32,28,35)(29,30,33,40)(36,39,38,37);(1,4)(2,7)(3,8)(5,6)(9,13,16,12)(10,19,15,18)(11,22,14,21)(17,24,20,23)(25,28,31,27)(26,32,34,35)(29,38,33,36)(30,39,40,37);(1,4)(2,3)(5,6)(7,8)(9,14,16,11)(10,17,15,20)(12,22,13,21)(18,23,19,24)(25,29)(26,30)(27,36)(28,38)(31,33)(32,39)(34,40)(35,37);(1,2)(3,6)(4,7)(5,8)(9,15,16,10)(11,20,14,17)(12,19,13,18)(21,24,22,23)(25,30)(26,33)(27,37)(28,39)(29,34)(31,40)(32,36)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9623447);