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 | |
4606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26,31,35)(27,30,37,39)(28,34,38,32)(29,33,36,40);(1,2)(3,7)(4,5)(6,8)(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(25,27,31,37)(26,30,35,39)(28,29,38,36)(32,40,34,33);(2,5)(6,7)(13,21)(16,23)(18,24)(20,22)(25,28,31,38)(26,32,35,34)(27,36,37,29)(30,33,39,40);(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,22)(23,24)(25,29)(26,33)(27,28)(30,34)(31,36)(32,39)(35,40)(37,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,30)(26,27)(28,33)(29,34)(31,39)(32,36)(35,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,17,23)(11,18,19,24)(12,20,15,22)(25,28)(26,34)(27,29)(30,33)(31,38)(32,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8105108);