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 | |
4431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,15)(12,16)(13,19)(14,17)(18,21)(20,23)(22,24)(25,26,31,35)(27,34,30,32)(28,29,33,40)(36,39,38,37);(1,3,6,5)(2,7,4,8)(9,11,14,19)(10,15,17,13)(12,18,22,20)(16,21,24,23);(9,11)(10,13)(12,18)(14,19)(15,17)(16,23)(20,22)(21,24)(25,27,31,30)(26,32,35,34)(28,36,33,38)(29,37,40,39);(1,2,6,4)(3,8,5,7)(9,11)(10,13)(12,20)(14,19)(15,17)(16,21)(18,22)(23,24)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(1,4,6,2)(3,8,5,7)(9,12)(10,16)(11,18)(13,23)(14,22)(15,21)(17,24)(19,20)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,38)(34,36);(1,5,6,3)(2,7,4,8)(9,13)(10,11)(12,21)(14,15)(16,20)(17,19)(18,24)(22,23)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,39,40,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10441570);