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 | #2682 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4111 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,18)(13,19)(14,16)(15,17)(21,24)(22,23)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(3,8)(6,7)(9,11,15,14)(10,16,17,20)(12,21,19,23)(13,24,18,22)(25,27,31,38)(26,32,36,37)(28,29,35,39)(30,34,33,40);(1,3)(2,6)(4,7)(5,8)(10,17)(12,19)(16,20)(21,23)(25,28,31,35)(26,33,36,30)(27,29,38,39)(32,40,37,34);(1,2,5,4)(3,7,8,6)(9,12,15,19)(10,18,17,13)(11,21,14,23)(16,22,20,24);(1,3)(2,7)(4,6)(5,8)(9,13,15,18)(10,19,17,12)(11,22,14,24)(16,21,20,23)(25,29)(26,34)(27,35)(28,38)(30,37)(31,39)(32,33)(36,40);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,20,17,16)(12,23,19,21)(13,22,18,24)(25,30,31,33)(26,35,36,28)(27,34,38,40)(29,32,39,37)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=h,e^2=hi,f^2=gi,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, 9620942);