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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,17)(11,16,14,18)(12,13,21,24)(19,20,22,23)(25,26,31,34)(27,28,29,32)(30,33,38,40)(35,36,37,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,14)(10,16,17,18)(12,19,21,22)(13,20,24,23)(25,27,31,29)(26,32,34,28)(30,37,38,35)(33,36,40,39);(10,17)(12,21)(16,18)(19,22)(25,28,31,32)(26,27,34,29)(30,36,38,39)(33,35,40,37);(9,12)(10,13)(11,19)(14,22)(15,21)(16,20)(17,24)(18,23)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,40)(38,39);(1,3,5,4)(2,7,6,8)(9,13)(10,12)(11,20)(14,23)(15,24)(16,19)(17,21)(18,22)(25,29,31,27)(26,28,34,32)(30,35,38,37)(33,39,40,36);(1,4,5,3)(2,8,6,7)(9,14,15,11)(10,18,17,16)(12,22,21,19)(13,23,24,20)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7880107);