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 | |
5040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26)(27,32)(28,38)(29,33)(30,34)(31,35)(36,40)(37,39);(2,6)(3,7)(9,11,13,19)(10,14,15,17)(12,18,21,20)(16,22,23,24)(25,27,31,36)(26,32,35,40)(28,39,33,30)(29,34,38,37);(1,3,5,7)(2,4,6,8)(10,15)(14,17)(16,23)(22,24)(27,36)(30,39)(32,40)(34,37);(1,4)(2,7)(3,6)(5,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28)(26,29)(27,30)(31,33)(32,37)(34,40)(35,38)(36,39);(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,29)(26,33)(27,37)(28,35)(30,40)(31,38)(32,39)(34,36);(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,30)(26,34)(27,28)(29,40)(31,39)(32,38)(33,36)(35,37)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8314095);