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 | |
4539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,39)(37,38);(1,2,5,4)(3,6,8,7)(10,11)(15,17)(16,18)(22,23)(25,27,31,28)(26,32,33,34)(29,36,40,38)(30,37,35,39);(2,4)(6,7)(13,21)(16,18)(19,24)(22,23)(26,33)(30,35)(32,34)(37,39);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,31,27)(26,34,33,32)(29,38,40,36)(30,39,35,37);(1,3,5,8)(2,6,4,7)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29)(26,35)(27,36)(28,38)(30,33)(31,40)(32,39)(34,37);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,35)(26,29,33,40)(27,37,28,39)(32,36,34,38)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8087624);