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 | |
4559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,14,15)(11,12,18,16)(13,21,22,17)(19,23,24,20)(25,26)(27,31)(28,32)(29,38)(30,34)(33,39)(35,36)(37,40);(2,5)(3,7)(10,15)(12,16)(13,22)(19,24)(25,27)(26,31)(28,29)(30,35)(32,38)(33,40)(34,36)(37,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(17,21)(19,24)(27,35)(28,37)(31,36)(32,40);(2,5)(6,8)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,28,30,37)(26,32,34,40)(27,29,35,39)(31,38,36,33);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29)(26,33)(27,28)(30,39)(31,40)(32,36)(34,38)(35,37);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^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, 8744542);