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 | #2703 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(17,20)(19,24)(27,35)(30,40)(32,36)(34,39);(1,2,4,6)(3,5,8,7)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,23)(21,24)(25,26,31,28)(27,36,35,32)(29,33,37,38)(30,34,40,39);(1,2)(3,5)(4,6)(7,8)(25,27,31,35)(26,32,28,36)(29,30,37,40)(33,39,38,34);(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,21,20,23)(25,28,31,26)(27,32,35,36)(29,38,37,33)(30,39,40,34);(1,2)(3,7)(4,6)(5,8)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,23,16,21)(25,29)(26,33)(27,30)(28,38)(31,37)(32,39)(34,36)(35,40);(1,3)(2,5)(4,8)(6,7)(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,19,20,24)(25,30)(26,34)(27,37)(28,39)(29,35)(31,40)(32,33)(36,38)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=gh,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, 9694544);