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 | |
4558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,19,22)(12,13,20,21)(17,18,23,24)(25,26)(27,36)(28,34)(29,39)(30,31)(32,35)(33,38)(37,40);(1,2,5,6)(3,7,8,4)(9,11,14,19)(10,15,16,22)(12,17,20,23)(13,18,21,24)(25,27)(26,33)(28,37)(29,30)(31,34)(32,38)(35,36)(39,40);(1,3)(2,4)(5,8)(6,7)(10,16)(12,20)(15,22)(17,23)(25,28,32,29)(26,34,35,39)(27,37,38,30)(31,36,40,33);(2,6)(4,7)(9,12)(10,13)(11,17)(14,20)(15,18)(16,21)(19,23)(22,24)(25,29,32,28)(26,34,35,39)(27,37,38,30)(31,33,40,36);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,30,32,37)(26,31,35,40)(27,29,38,28)(33,34,36,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,12)(11,18)(14,21)(15,17)(16,20)(19,24)(22,23)(25,31)(26,30)(27,34)(28,36)(29,33)(32,40)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=g,e^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, 10132789);