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 | |
4663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,14,15)(11,18,17,13)(12,20,21,16)(19,23,22,24)(27,33)(30,34)(35,39)(38,40);(1,2,4,6)(3,5,7,8)(10,15)(12,21)(13,18)(19,22)(25,26)(27,30)(28,36)(29,32)(31,37)(33,34)(35,40)(38,39);(1,3)(2,5)(4,7)(6,8)(10,15)(11,17)(16,20)(19,22)(25,26,29,32)(27,30,33,34)(28,31,37,36)(35,38,39,40);(9,11,14,17)(10,13,15,18)(12,22,21,19)(16,24,20,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,23)(14,21)(15,20)(17,22)(18,24)(25,27)(26,30)(28,35)(29,33)(31,38)(32,34)(36,40)(37,39);(1,3)(2,5)(4,7)(6,8)(9,13)(10,17)(11,15)(12,23)(14,18)(16,22)(19,20)(21,24)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8736216);