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 | |
4342 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)(17,24)(19,20)(25,26)(27,34)(28,32)(29,37)(30,31)(33,39)(35,36)(38,40);(1,2)(3,6)(4,7)(5,8)(9,10,14,15)(11,16,18,13)(12,17,22,24)(19,21,20,23)(26,31)(28,34)(33,37)(36,38);(1,3)(2,5)(4,8)(6,7)(25,27,30,32)(26,28,31,34)(29,35,39,40)(33,36,37,38);(1,2)(3,5)(4,7)(6,8)(9,11)(10,16)(12,20)(13,15)(14,18)(17,23)(19,22)(21,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,17,15,24)(11,19,18,20)(13,23,16,21)(25,28)(26,32)(27,31)(29,38)(30,34)(33,35)(36,39)(37,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,20,24,19)(25,29,30,39)(26,33,31,37)(27,35,32,40)(28,36,34,38)Polycyclic presentation:
2^9|b^2=gh,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9995026);