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 | |
4678 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)(11,20)(12,18)(13,22)(14,15)(16,19)(17,21)(23,24)(25,26)(27,34)(28,36)(29,32)(30,33)(31,38)(35,39)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11,16,21)(10,17,19,20)(12,15,13,23)(14,18,24,22)(25,26)(27,34)(28,31)(29,32)(30,33)(35,37)(36,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12,16,13)(10,18,19,22)(11,15,21,23)(14,17,24,20)(25,27,30,32)(26,29,33,34)(28,35,38,40)(31,39,36,37);(2,6)(5,7)(9,13,16,12)(10,18,19,22)(11,15,21,23)(14,20,24,17)(25,26)(27,29)(28,31)(30,33)(32,34)(35,39)(36,38)(37,40);(1,3)(2,7)(4,8)(5,6)(9,14,16,24)(10,15,19,23)(11,18,21,22)(12,17,13,20)(25,28)(26,31)(27,35)(29,39)(30,38)(32,40)(33,36)(34,37);(1,2)(3,5)(4,6)(7,8)(9,15)(10,14)(11,13)(12,21)(16,23)(17,18)(19,24)(20,22)(25,29)(26,32)(27,33)(28,37)(30,34)(31,35)(36,40)(38,39)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8596244);