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 | |
4279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,14,17)(11,15,18,12)(13,16,22,24)(19,23,21,20)(25,26)(27,29)(28,36)(30,31)(32,34)(33,38)(35,39)(37,40);(2,5)(3,7)(9,11,14,18)(10,15,17,12)(13,21,22,19)(16,20,24,23)(26,31)(27,34)(33,36)(35,40);(1,2)(3,6)(4,5)(7,8)(11,18)(12,15)(19,21)(20,23)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(2,5)(6,8)(9,12)(10,11)(13,20)(14,15)(16,19)(17,18)(21,24)(22,23)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,21)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=g,f^2=h,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, 8735863);