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 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(13,21)(17,22)(18,23)(19,24)(25,26,31,34)(27,29,32,28)(30,38,39,33)(35,36,40,37);(3,7)(6,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,27,31,32)(26,28,34,29)(30,35,39,40)(33,36,38,37);(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(17,22)(18,23)(25,28,31,29)(26,32,34,27)(30,36,39,37)(33,40,38,35);(1,3,4,7)(2,6,5,8)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,29,31,28)(26,27,34,32)(30,37,39,36)(33,35,38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=ghi,f^2=gi,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, 8022731);