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 | #57354 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,20)(11,21,22,16)(12,17,19,14)(13,18,23,24)(25,26)(27,36)(28,29)(30,38)(31,32)(33,37)(34,40)(35,39);(1,2)(3,6)(4,7)(5,8)(11,22)(13,23)(16,21)(18,24)(26,32)(27,37)(29,34)(35,38);(1,3)(2,5)(4,8)(6,7)(9,11,15,22)(10,16,20,21)(12,23,19,13)(14,24,17,18)(25,27)(26,33)(28,38)(29,39)(30,34)(31,37)(32,36)(35,40);(3,8)(5,6)(9,12,15,19)(10,17,20,14)(11,13,22,23)(16,24,21,18)(25,28)(26,34)(27,38)(29,32)(30,33)(31,40)(35,37)(36,39);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,12)(14,16)(15,23)(17,21)(19,22)(20,24)(25,29,31,34)(26,28,32,40)(27,39,37,30)(33,38,36,35);(1,2)(3,6)(4,7)(5,8)(9,14,15,17)(10,19,20,12)(11,18,22,24)(13,16,23,21)(25,30,31,39)(26,35,32,38)(27,34,37,29)(28,33,40,36)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8811775);