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 | #41007 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4225 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)(25,26,30,31)(27,34,29,32)(28,33,36,38)(35,40,37,39);(3,4)(7,8)(9,10,15,19)(11,20,22,16)(12,14,18,24)(13,17,21,23)(26,31)(28,36)(32,34)(35,37);(1,3)(2,7)(4,6)(5,8)(9,11,15,22)(10,16,19,20)(12,13,18,21)(14,23,24,17)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(3,4)(7,8)(9,12,15,18)(10,14,19,24)(11,21,22,13)(16,17,20,23)(25,27,30,29)(26,32,31,34)(28,35,36,37)(33,39,38,40);(1,4,6,3)(2,7,5,8)(9,13,15,21)(10,17,19,23)(11,12,22,18)(14,16,24,20)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,5,6,2)(3,7,4,8)(9,14)(10,18)(11,17)(12,19)(13,16)(15,24)(20,21)(22,23)(25,29,30,27)(26,34,31,32)(28,37,36,35)(33,40,38,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=hi,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, 8923845);