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 | |
4531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,14,15)(11,16,20,23)(12,19,21,17)(13,18,24,22)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40);(1,2,4,6)(3,5,7,8)(10,15)(12,21)(16,23)(18,22)(25,26)(27,31)(28,36)(29,30)(32,38)(33,35)(34,40)(37,39);(1,3)(2,5)(4,7)(6,8)(10,15)(13,24)(16,23)(17,19)(26,30)(31,33)(32,36)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,23)(17,24)(18,21)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,3)(2,5)(4,7)(6,8)(9,12,14,21)(10,17,15,19)(11,18,20,22)(13,23,24,16)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37);(9,13)(10,18)(11,19)(12,23)(14,24)(15,22)(16,21)(17,20)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8313713);