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 | |
4479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,5,7)(9,10,14,16)(11,15,18,12)(13,17,22,23)(19,21,24,20)(26,31)(27,33)(32,35)(34,39);(1,3,6,5)(2,7,8,4)(9,11)(10,15)(12,16)(13,19)(14,18)(17,21)(20,23)(22,24)(25,26,30,31)(27,29,33,38)(28,35,36,32)(34,40,39,37);(2,8)(4,7)(10,16)(11,18)(17,23)(19,24)(28,36)(32,35)(34,39)(37,40);(2,8)(3,5)(9,12)(10,11)(13,21)(14,15)(16,18)(17,24)(19,23)(20,22)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,40)(36,39);(1,4,6,7)(2,3,8,5)(9,10)(11,12)(13,17)(14,16)(15,18)(19,20)(21,24)(22,23)(25,28,30,36)(26,32,31,35)(27,34,33,39)(29,37,38,40);(1,5,6,3)(2,7,8,4)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29)(26,33)(27,31)(28,37)(30,38)(32,39)(34,35)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8540731);