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 | |
4438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,18,16,12)(13,17,22,23)(19,24,21,20)(25,26,30,34)(27,31,37,32)(28,29,33,39)(35,40,38,36);(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(17,23)(19,21)(25,27)(26,31)(28,38)(29,36)(30,37)(32,34)(33,35)(39,40);(2,5)(6,8)(10,15)(12,18)(17,23)(20,24)(25,27,30,37)(26,32,34,31)(28,35,33,38)(29,36,39,40);(3,7)(6,8)(9,11,14,16)(10,12,15,18)(13,21,22,19)(17,24,23,20)(25,28)(26,29)(27,35)(30,33)(31,40)(32,36)(34,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,21)(19,23)(22,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,34)(33,38)(39,40);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)(25,29)(26,33)(27,36)(28,34)(30,39)(31,35)(32,38)(37,40)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8451009);