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 | |
4784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,16)(12,22)(13,18)(14,15)(17,21)(19,24)(20,23)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(2,5)(3,7)(10,15)(13,22)(16,17)(20,24)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,16,15,17)(12,19,18,23)(13,20,22,24)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(3,7)(6,8)(9,11)(10,17)(12,23)(13,20)(14,21)(15,16)(18,19)(22,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,18)(10,13,15,22)(11,19,21,23)(16,20,17,24)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,20)(12,15)(14,22)(16,23)(17,19)(21,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,35)(34,40)(38,39)Polycyclic presentation:
2^9|b^2=hi,c^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297714);