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 | #57180 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,15)(13,17)(14,16)(19,24)(20,22)(21,23)(25,26,31,36)(27,37,40,32)(28,33,39,34)(29,35,38,30);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(25,27)(26,32)(28,38)(29,39)(30,34)(31,40)(33,35)(36,37);(1,3)(2,4)(5,6)(7,8)(10,12)(16,18)(17,20)(23,24)(25,28,31,39)(26,33,36,34)(27,38,40,29)(30,32,35,37);(3,6)(4,8)(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,28)(26,34)(27,38)(29,40)(30,32)(31,39)(33,36)(35,37);(1,4,5,8)(2,6,7,3)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29,31,38)(26,35,36,30)(27,39,40,28)(32,33,37,34);(1,2)(3,4)(5,7)(6,8)(9,14,15,11)(10,18,12,16)(13,19,22,21)(17,23,20,24)(25,30,31,35)(26,29,36,38)(27,34,40,33)(28,32,39,37)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068277);