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 | |
4343 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)(4,5)(6,8)(9,10,14,15)(11,19,20,16)(12,17,18,13)(21,24,23,22)(25,26)(27,30)(28,35)(29,31)(32,36)(33,39)(34,37)(38,40);(3,8)(6,7)(10,15)(12,18)(16,19)(21,23)(25,27,29,34)(26,30,31,37)(28,33,36,40)(32,38,35,39);(1,3)(2,6)(4,7)(5,8)(10,15)(11,20)(13,17)(21,23)(26,31)(30,37)(32,35)(38,39);(3,8)(6,7)(9,11,14,20)(10,16,15,19)(12,23,18,21)(13,22,17,24)(25,26)(27,30)(28,32)(29,31)(33,38)(34,37)(35,36)(39,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,18)(10,17,15,13)(11,21,20,23)(16,22,19,24)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,4,5,2)(3,6,8,7)(9,13,14,17)(10,18,15,12)(11,22,20,24)(16,21,19,23)(25,28,29,36)(26,32,31,35)(27,33,34,40)(30,38,37,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=g,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=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9347765);