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 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,18,13,15)(12,16,20,23)(19,24,21,22)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2)(3,7)(4,6)(5,8)(11,13)(12,20)(15,18)(16,23)(25,27,31,38)(26,32,36,37)(28,29,35,39)(30,34,33,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,15,17,18)(12,21,20,19)(16,24,23,22)(25,28,31,35)(26,33,36,30)(27,29,38,39)(32,40,37,34);(2,6)(5,7)(9,12,14,20)(10,16,17,23)(11,19,13,21)(15,22,18,24)(25,27)(26,32)(28,29)(30,34)(31,38)(33,40)(35,39)(36,37);(1,2)(3,5)(4,6)(7,8)(9,13,14,11)(10,15,17,18)(12,19,20,21)(16,24,23,22)(25,29)(26,34)(27,35)(28,38)(30,37)(31,39)(32,33)(36,40);(1,3)(2,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,23)(21,24)(25,30,31,33)(26,35,36,28)(27,34,38,40)(29,32,39,37)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10126766);