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 | |
4415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,21)(14,15)(16,18)(17,22)(19,23)(20,24)(26,32)(28,37)(33,34)(35,38);(1,2)(3,6)(4,5)(7,8)(10,15)(12,16)(17,21)(20,23)(25,26)(27,33)(28,30)(29,34)(31,32)(35,39)(36,38)(37,40);(2,5)(6,8)(10,15)(11,18)(13,22)(20,23)(25,27,31,29)(26,33,32,34)(28,35,37,38)(30,39,40,36);(3,7)(6,8)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,28)(26,30)(27,35)(29,38)(31,37)(32,40)(33,39)(34,36);(1,3)(2,6)(4,7)(5,8)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29,31,27)(26,34,32,33)(28,35,37,38)(30,39,40,36);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,30)(26,28)(27,36)(29,39)(31,40)(32,37)(33,38)(34,35)Polycyclic presentation:
2^9|c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9341814);