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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(25,26,31,36)(27,37,39,32)(28,35,30,33)(29,34,38,40);(3,4)(7,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,26)(27,37)(28,33)(29,40)(30,35)(31,36)(32,39)(34,38);(1,3)(2,7)(4,6)(5,8)(10,11)(12,19)(13,21)(22,23)(25,27,31,39)(26,32,36,37)(28,38,30,29)(33,40,35,34);(3,4)(7,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,31,30)(26,33,36,35)(27,38,39,29)(32,40,37,34);(1,4,6,3)(2,7,5,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29)(26,34)(27,28)(30,39)(31,38)(32,33)(35,37)(36,40);(1,5,6,2)(3,7,4,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,28)(26,35,36,33)(27,29,39,38)(32,34,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8919209);