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 | #43882 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4265 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(25,26,31,35)(27,36,39,32)(28,29,40,33)(30,34,38,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,19)(20,24)(21,22)(25,27,31,39)(26,32,35,36)(28,37,40,34)(29,38,33,30);(10,13)(15,18)(16,21)(23,24)(25,28,31,40)(26,33,35,29)(27,37,39,34)(30,36,38,32);(9,11,14,19)(10,15,13,18)(12,20,22,17)(16,24,21,23)(25,28)(26,29)(27,37)(30,36)(31,40)(32,38)(33,35)(34,39);(9,12,14,22)(10,16,13,21)(11,17,19,20)(15,23,18,24)(25,29)(26,28)(27,38)(30,39)(31,33)(32,37)(34,36)(35,40);(1,3,4,7)(2,5,6,8)(9,13,14,10)(11,18,19,15)(12,21,22,16)(17,24,20,23)(25,30,31,38)(26,34,35,37)(27,29,39,33)(28,36,40,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=i,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7756225);