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 | |
4565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,13)(12,20)(14,16)(15,19)(17,21)(18,23)(22,24)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(1,2)(3,7)(4,5)(6,8)(9,11,14,19)(10,15,16,13)(12,18,21,24)(17,23,20,22)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(2,5)(6,7)(10,16)(12,21)(13,15)(18,24)(25,26,29,32)(27,33,35,30)(28,31,38,36)(34,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,18)(13,22)(14,21)(15,23)(16,20)(19,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,15)(10,11,16,19)(12,22,21,23)(17,18,20,24)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,40)(37,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,16)(14,19)(17,23)(20,22)(21,24)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37)Polycyclic presentation:
2^9|b^2=h,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8665245);