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 | |
4577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(26,31)(27,34)(33,40)(35,36);(1,3,7,6)(2,4,8,5)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,26,30,31)(27,32,34,29)(28,33,38,40)(35,37,36,39);(1,4,7,5)(2,3,8,6)(10,15)(11,18)(16,23)(19,20);(2,8)(4,5)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,27)(26,32)(28,36)(29,31)(30,34)(33,39)(35,38)(37,40);(1,5,7,4)(2,3,8,6)(9,12,14,22)(10,16,15,23)(11,19,18,20)(13,21,17,24)(25,28,30,38)(26,33,31,40)(27,35,34,36)(29,39,32,37);(1,6,7,3)(2,5,8,4)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,36,40,35)Polycyclic presentation:
2^9|b^2=gi,c^2=hi,e^2=g,f^2=gi,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, 10103281);