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 | |
4468 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)(28,35)(29,31)(32,37)(33,39)(34,40)(36,38);(1,2)(3,7)(4,5)(6,8)(10,15)(12,20)(13,18)(19,23)(25,27,29,33)(26,30,31,39)(28,36,37,34)(32,40,35,38);(2,5)(6,7)(10,15)(11,17)(12,20)(21,24)(26,31)(27,33)(32,35)(34,36);(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)(28,32)(29,31)(33,39)(34,38)(35,37)(36,40);(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)(29,33)(31,39)(32,38)(35,40)(36,37);(1,3)(2,6)(4,8)(5,7)(9,13)(10,17)(11,15)(12,21)(14,18)(16,23)(19,22)(20,24)(25,28,29,37)(26,32,31,35)(27,34,33,36)(30,38,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=g,f^2=gh,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, 8667040);