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 | |
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)(25,26,31,32)(27,30,37,33)(28,36,39,34)(29,35,40,38);(1,2,5,7)(3,4,8,6)(9,11,15,12)(10,16,17,18)(13,22,19,20)(14,23,24,21)(26,32)(29,40)(30,33)(34,36);(1,3,5,8)(2,6,7,4)(10,17)(11,12)(13,19)(21,23)(25,27)(26,33)(28,38)(29,34)(30,32)(31,37)(35,39)(36,40);(9,12,15,11)(10,18,17,16)(13,22,19,20)(14,23,24,21)(25,28,31,39)(26,34,32,36)(27,35,37,38)(29,33,40,30);(1,3)(2,4)(5,8)(6,7)(9,13,15,19)(10,14,17,24)(11,20,12,22)(16,21,18,23)(25,29)(26,35)(27,36)(28,33)(30,39)(31,40)(32,38)(34,37);(1,4)(2,3)(5,6)(7,8)(9,14)(10,19)(11,21)(12,23)(13,17)(15,24)(16,22)(18,20)(25,30,31,33)(26,27,32,37)(28,29,39,40)(34,35,36,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=ghi,d^2=h,e^2=gh,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, 9157487);