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 | #2407 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,15,19)(11,20,14,16)(12,22,23,17)(13,18,24,21)(25,26)(27,37)(28,39)(29,35)(30,31)(32,36)(33,40)(34,38);(2,6)(3,7)(9,11,15,14)(10,16,19,20)(12,18,23,21)(13,17,24,22)(25,27)(26,33)(28,29)(30,38)(31,39)(32,40)(34,35)(36,37);(1,3)(2,4)(5,7)(6,8)(25,28,32,38)(26,34,36,39)(27,29,40,30)(31,33,35,37);(1,4)(2,3)(5,8)(6,7)(9,12,15,23)(10,17,19,22)(11,21,14,18)(13,20,24,16)(25,29,32,30)(26,31,36,35)(27,28,40,38)(33,39,37,34);(9,13)(10,18)(11,17)(12,20)(14,22)(15,24)(16,23)(19,21)(25,30,32,29)(26,35,36,31)(27,38,40,28)(33,34,37,39);(9,14,15,11)(10,16,19,20)(12,21,23,18)(13,17,24,22)(25,31,32,35)(26,30,36,29)(27,39,40,34)(28,33,38,37)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8608684);