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 |
| prime radical | #2617 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4338 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)(3,5,4,7)(13,21)(16,22)(18,23)(19,24)(25,26,31,34)(27,35,38,32)(28,39,37,29)(30,36,40,33);(1,3,6,4)(2,7,8,5)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,27,31,38)(26,32,34,35)(28,30,37,40)(29,36,39,33);(1,3,6,4)(2,7,8,5)(10,15)(11,17)(16,22)(18,23)(25,26,31,34)(27,32,38,35)(28,39,37,29)(30,33,40,36);(2,8)(3,4)(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(25,28,31,37)(26,29,34,39)(27,30,38,40)(32,36,35,33);(1,4,6,3)(2,5,8,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29)(26,28)(27,36)(30,32)(31,39)(33,38)(34,37)(35,40);(1,5)(2,3)(4,8)(6,7)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30)(26,33)(27,37)(28,38)(29,32)(31,40)(34,36)(35,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8067040);