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 | |
4431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(25,26,30,31)(27,32,36,33)(28,37,39,29)(34,35,38,40);(1,2)(3,7)(4,5)(6,8)(10,15)(12,17)(16,22)(20,23)(25,26,30,31)(27,33,36,32)(28,37,39,29)(34,40,38,35);(2,5)(6,7)(13,21)(16,22)(18,24)(20,23)(26,31)(29,37)(32,33)(35,40);(3,8)(6,7)(9,11,14,19)(10,12,15,17)(13,18,21,24)(16,20,22,23)(25,27,30,36)(26,32,31,33)(28,38,39,34)(29,35,37,40);(1,3,4,8)(2,6,5,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,28)(26,29)(27,34)(30,39)(31,37)(32,40)(33,35)(36,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,29)(26,28)(27,35)(30,37)(31,39)(32,38)(33,34)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=gh,e^2=hi,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=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8269272);