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 | #37737 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4211 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)(11,16)(12,18)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26,32,30)(27,28,37,33)(29,39,34,31)(35,40,38,36);(1,2,5,7)(3,4,8,6)(9,11,15,12)(10,16,17,18)(13,22,19,20)(14,23,24,21)(25,27,32,37)(26,33,30,28)(29,35,34,38)(31,40,39,36);(1,3,5,8)(2,6,7,4)(10,17)(11,12)(13,19)(21,23)(25,28)(26,27)(29,40)(30,37)(31,38)(32,33)(34,36)(35,39);(9,12,15,11)(10,18,17,16)(13,22,19,20)(14,23,24,21)(25,29)(26,31)(27,35)(28,36)(30,39)(32,34)(33,40)(37,38);(1,3)(2,4)(5,8)(6,7)(9,13,15,19)(10,14,17,24)(11,20,12,22)(16,21,18,23)(25,30,32,26)(27,33,37,28)(29,39,34,31)(35,40,38,36);(1,4)(2,3)(5,6)(7,8)(9,14)(10,19)(11,21)(12,23)(13,17)(15,24)(16,22)(18,20)(25,31,32,39)(26,34,30,29)(27,36,37,40)(28,38,33,35)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=ghi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9157481);