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 | |
4461 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)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,35)(27,36,34,30)(28,39,38,32)(29,33,37,40);(3,8)(5,6)(12,19)(16,22)(18,23)(20,24)(25,27,31,34)(26,30,35,36)(28,37,38,29)(32,40,39,33);(1,3)(2,5)(4,8)(6,7)(10,15)(11,17)(16,22)(18,23)(25,28,31,38)(26,32,35,39)(27,37,34,29)(30,40,36,33);(3,8)(5,6)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28)(26,32)(27,37)(29,34)(30,40)(31,38)(33,36)(35,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,37)(26,33,35,40)(27,38,34,28)(30,39,36,32);(1,2)(3,5)(4,7)(6,8)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30)(26,34)(27,35)(28,40)(29,32)(31,36)(33,38)(37,39)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10028342);