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 | |
4255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,18,16)(12,17,22,24)(19,23,20,21)(25,26,31,34)(27,28,29,32)(30,38,39,33)(35,40,37,36);(1,2)(3,7)(4,6)(5,8)(10,15)(11,18)(17,24)(19,20)(25,27)(26,32)(28,34)(29,31)(30,35)(33,40)(36,38)(37,39);(1,3)(2,5)(4,8)(6,7)(25,28,31,32)(26,27,34,29)(30,36,39,40)(33,35,38,37);(2,6)(5,7)(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,23,24,21)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,22)(10,17,15,24)(11,19,18,20)(13,23,16,21)(25,29,31,27)(26,32,34,28)(30,35,39,37)(33,36,38,40);(1,3)(2,7)(4,8)(5,6)(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,19,24,20)(25,30,31,39)(26,33,34,38)(27,35,29,37)(28,36,32,40)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gh,e^2=g,f^2=g,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, 10125717);