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 | #39838 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4123 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)(3,7)(4,6)(5,8)(9,10,14,17)(11,13,19,15)(12,16,20,23)(18,21,24,22)(25,26,30,33)(27,28,31,36)(29,32,38,40)(34,35,37,39);(1,2,4,6)(3,5,8,7)(12,20)(16,23)(18,24)(21,22)(25,27,30,31)(26,28,33,36)(29,37,38,34)(32,39,40,35);(1,2)(3,5)(4,6)(7,8)(9,11,14,19)(10,15,17,13)(12,18,20,24)(16,22,23,21)(25,27)(26,28)(29,34)(30,31)(32,35)(33,36)(37,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,24)(22,23);(9,12,14,20)(10,16,17,23)(11,18,19,24)(13,21,15,22)(25,28)(26,31)(27,33)(29,35)(30,36)(32,37)(34,40)(38,39);(9,13)(10,11)(12,21)(14,15)(16,18)(17,19)(20,22)(23,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150963);