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 | |
4672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,12)(13,16,22,23)(19,21,24,20)(25,26,31,33)(27,30,28,32)(29,34,38,40)(35,37,36,39);(1,2)(3,7)(4,6)(5,8)(10,15)(11,18)(16,23)(19,24)(25,27,31,28)(26,32,33,30)(29,36,38,35)(34,37,40,39);(1,3)(2,5)(4,8)(6,7)(26,33)(27,28)(34,40)(35,36);(2,6)(5,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,23,21)(25,28,31,27)(26,30,33,32)(29,36,38,35)(34,37,40,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,21)(14,17)(15,18)(16,24)(19,23)(20,22)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40);(1,3)(2,7)(4,8)(5,6)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,30)(26,27)(28,33)(29,37)(31,32)(34,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^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, 10125039);