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 | |
4573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,18)(16,22)(17,19)(20,24)(21,23)(25,26)(27,35)(28,37)(29,34)(30,38)(31,33)(32,36)(39,40);(1,3)(2,6)(4,7)(5,8)(10,11)(12,19)(16,18)(20,21)(25,27,31,36)(26,32,33,35)(28,29,38,39)(30,34,37,40);(10,11)(15,17)(16,18)(23,24)(26,33)(27,36)(28,38)(34,40);(1,3,5,8)(2,6,4,7)(9,11,14,10)(12,15,19,17)(13,18,22,16)(20,23,21,24)(25,28)(26,30)(27,29)(31,38)(32,34)(33,37)(35,40)(36,39);(1,3)(2,7)(4,6)(5,8)(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,29)(26,34)(27,28)(30,32)(31,39)(33,40)(35,37)(36,38);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,30)(26,28)(27,34)(29,32)(31,37)(33,38)(35,39)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154609);