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 | |
4703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,16)(11,12,18,15)(13,17,22,24)(19,20,21,23)(25,26,31,33)(27,35,29,32)(28,34,38,30)(36,37,39,40);(1,2)(3,6)(4,7)(5,8)(9,11,14,18)(10,15,16,12)(13,21,22,19)(17,20,24,23)(25,27,31,29)(26,32,33,35)(28,36,38,39)(30,37,34,40);(10,16)(11,18)(13,22)(20,23)(26,33)(28,38)(32,35)(36,39);(9,11)(10,15)(12,16)(13,19)(14,18)(17,23)(20,24)(21,22)(25,28)(26,34)(27,36)(29,39)(30,33)(31,38)(32,40)(35,37);(1,3,5,4)(2,6,8,7)(9,12,14,15)(10,11,16,18)(13,20,22,23)(17,19,24,21)(25,29,31,27)(26,35,33,32)(28,39,38,36)(30,40,34,37);(1,4,5,3)(2,7,8,6)(9,13,14,22)(10,17,16,24)(11,19,18,21)(12,20,15,23)(25,30,31,34)(26,28,33,38)(27,37,29,40)(32,36,35,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713920);