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 |
| prime radical | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,31,32)(27,29,34,39)(28,33,37,40)(30,38,36,35);(1,2,5,4)(3,7,8,6)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27)(26,29)(28,36)(30,37)(31,34)(32,39)(33,35)(38,40);(1,3,5,8)(2,6,4,7)(26,32)(27,34)(28,37)(35,38);(1,4,5,2)(3,7,8,6)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,28,31,37)(26,33,32,40)(27,36,34,30)(29,35,39,38);(9,13,14,10)(11,17,19,15)(12,16,21,20)(18,22,24,23)(25,29)(26,34)(27,32)(28,38)(30,40)(31,39)(33,36)(35,37);(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,22)(21,24)(25,30,31,36)(26,35,32,38)(27,37,34,28)(29,33,39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=hi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10073471);