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 | |
4406 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)(13,21)(17,22)(18,23)(19,24)(26,28)(30,38)(32,35)(34,37);(1,2,4,5)(3,6,7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,31,28)(27,32,29,35)(30,33,38,36)(34,39,37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(17,22)(18,23)(25,27,31,29)(26,32,28,35)(30,34,38,37)(33,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,28,31,26)(27,32,29,35)(30,36,38,33)(34,39,37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29,31,27)(26,32,28,35)(30,37,38,34)(33,39,36,40);(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019743);