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 | #36316 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18)(11,19,21,15)(12,16,22,20)(13,17,23,24)(25,26)(27,34)(28,36)(29,31)(30,33)(32,37)(35,39)(38,40);(1,2,4,6)(3,5,8,7)(9,11,14,21)(10,15,18,19)(12,17,22,24)(13,20,23,16)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,37)(38,39);(1,2)(3,5)(4,6)(7,8)(25,27,30,29)(26,31,33,34)(28,35,37,38)(32,39,36,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,17)(13,16)(14,21)(18,19)(20,23)(22,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,37)(38,40);(9,12)(10,16)(11,17)(13,15)(14,22)(18,20)(19,23)(21,24)(25,28)(26,32)(27,35)(29,38)(30,37)(31,39)(33,36)(34,40);(9,13)(10,17)(11,20)(12,19)(14,23)(15,22)(16,21)(18,24)(25,29,30,27)(26,31,33,34)(28,35,37,38)(32,40,36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8599098);