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 | #56717 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,21,22,16)(12,13,19,18)(14,24,23,20)(26,32)(28,37)(33,39)(35,38);(2,5)(6,7)(9,11)(10,16)(12,23)(13,24)(14,19)(15,22)(17,21)(18,20)(25,26,31,32)(27,33,29,39)(28,34,37,30)(35,40,38,36);(1,2)(3,6)(4,5)(7,8)(10,17)(11,22)(12,19)(20,24)(25,27,31,29)(26,33,32,39)(28,35,37,38)(30,36,34,40);(9,12,15,19)(10,18,17,13)(11,23,22,14)(16,20,21,24)(25,28)(26,34)(27,35)(29,38)(30,32)(31,37)(33,40)(36,39);(1,2)(3,7)(4,5)(6,8)(9,13,15,18)(10,19,17,12)(11,20,22,24)(14,16,23,21)(25,29,31,27)(26,33,32,39)(28,38,37,35)(30,36,34,40);(1,3)(2,6)(4,8)(5,7)(9,14,15,23)(10,20,17,24)(11,19,22,12)(13,16,18,21)(25,30,31,34)(26,28,32,37)(27,36,29,40)(33,35,39,38)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9686687);