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 | |
4743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,30,33)(27,31,35,29)(28,32,36,39)(34,37,40,38);(3,8)(5,6)(12,19)(16,22)(18,23)(20,24)(25,27,30,35)(26,29,33,31)(28,34,36,40)(32,38,39,37);(1,3,4,8)(2,5,7,6)(10,15)(11,17)(16,22)(18,23)(28,36)(32,39)(34,40)(37,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,37)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,28)(26,32)(27,34)(29,38)(30,36)(31,37)(33,39)(35,40);(1,2)(3,5)(4,7)(6,8)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,29,30,31)(26,27,33,35)(28,37,36,38)(32,40,39,34)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9965705);