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 | |
4388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,24)(21,23)(25,26)(27,29)(28,36)(30,31)(32,39)(33,34)(35,40)(37,38);(2,5)(6,7)(10,15)(12,17)(16,18)(20,23)(25,27)(26,29)(28,37)(30,33)(31,34)(32,40)(35,39)(36,38);(1,2)(3,6)(4,5)(7,8)(11,19)(16,18)(20,23)(21,24)(26,31)(27,33)(32,36)(35,37);(1,3)(2,6)(4,8)(5,7)(9,11,14,19)(10,16,15,18)(12,20,17,23)(13,21,22,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(9,12,14,17)(10,13,15,22)(11,20,19,23)(16,21,18,24)(25,28,30,39)(26,32,31,36)(27,35,33,37)(29,38,34,40);(9,13)(10,17)(11,21)(12,15)(14,22)(16,23)(18,20)(19,24)(25,29,30,34)(26,33,31,27)(28,38,39,40)(32,37,36,35)Polycyclic presentation:
2^9|d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021623);