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 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(5,6)(9,10)(11,13)(12,15)(14,20)(16,18)(17,22)(19,23)(21,24);(1,2,4,7)(3,6,8,5)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(25,26,30,31)(27,29,33,38)(28,32,35,39)(34,37,36,40);(1,3,4,8)(2,5,7,6)(10,12)(11,18)(17,20)(19,24)(26,31)(28,35)(29,38)(34,36);(1,2,4,7)(3,6,8,5)(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,27,30,33)(26,29,31,38)(28,36,35,34)(32,40,39,37);(1,2)(3,6)(4,7)(5,8)(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40);(1,3)(2,6)(4,8)(5,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29)(26,33)(27,31)(28,37)(30,38)(32,36)(34,39)(35,40)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9414108);