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 | #57529 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(12,20)(15,17)(18,21)(25,26,31,36)(27,33,29,32)(28,38,39,34)(30,35,40,37);(1,2)(3,7)(4,6)(5,8)(9,10,13,14)(11,17,19,15)(12,16,20,22)(18,24,21,23)(25,27,31,29)(26,32,36,33)(28,35,39,37)(30,34,40,38);(1,3)(2,5)(4,8)(6,7)(9,10,13,14)(11,15,19,17)(12,16,20,22)(18,23,21,24)(25,27,31,29)(26,33,36,32)(28,37,39,35)(30,34,40,38);(2,6)(5,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,28)(26,34)(27,37)(29,35)(30,32)(31,39)(33,40)(36,38);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,15,14,17)(12,21,20,18)(16,24,22,23)(25,29,31,27)(26,32,36,33)(28,37,39,35)(30,34,40,38);(1,3)(2,7)(4,8)(5,6)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,24)(19,21)(25,30,31,40)(26,35,36,37)(27,34,29,38)(28,32,39,33)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10127032);