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 | |
4511 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)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,2)(3,4)(5,6)(7,8)(10,11)(12,19)(16,18)(20,24)(25,27,31,28)(26,32,33,34)(29,38,40,36)(30,37,35,39);(2,6)(4,7)(13,21)(16,18)(20,24)(22,23)(26,33)(30,35)(32,34)(37,39);(2,6)(3,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,31,27)(26,34,33,32)(29,36,40,38)(30,39,35,37);(1,3)(2,7)(4,6)(5,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29)(26,35)(27,36)(28,38)(30,33)(31,40)(32,37)(34,39);(1,4,5,7)(2,3,6,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,35)(26,29,33,40)(27,37,28,39)(32,38,34,36)Polycyclic presentation:
2^9|b^2=h,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306223);