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 | |
4418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,18,17,13)(12,16,20,22)(19,24,23,21)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(1,2)(3,7)(4,5)(6,8)(10,15)(12,20)(13,18)(19,23)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(2,5)(6,7)(10,15)(11,17)(12,20)(21,24)(25,26,29,32)(27,33,35,30)(28,31,38,36)(34,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,17)(10,13,15,18)(12,19,20,23)(16,21,22,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,40)(37,38);(1,3)(2,6)(4,8)(5,7)(9,13)(10,17)(11,15)(12,21)(14,18)(16,23)(19,22)(20,24)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^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, 8665240);