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 | |
4632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(17,22)(19,24)(25,26)(27,36)(28,30)(29,39)(31,33)(32,40)(34,38)(35,37);(1,2,5,6)(3,7,8,4)(9,10,14,15)(11,12,18,16)(13,17,21,22)(19,20,24,23)(25,27,31,40)(26,32,33,36)(28,37,38,29)(30,39,34,35);(2,6)(4,7)(13,21)(17,22)(19,24)(20,23)(26,33)(30,34)(32,36)(35,39);(2,6)(3,8)(9,11,14,18)(10,16,15,12)(13,19,21,24)(17,23,22,20)(25,28)(26,34)(27,37)(29,40)(30,33)(31,38)(32,35)(36,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,16)(10,11,15,18)(13,20,21,23)(17,19,22,24)(25,29,31,37)(26,35,33,39)(27,38,40,28)(30,32,34,36);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,34)(26,28,33,38)(27,39,40,35)(29,32,37,36)Polycyclic presentation:
2^9|b^2=gi,d^2=gh,e^2=g,f^2=h,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, 8296169);