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 | |
4332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(27,34)(29,31)(35,40)(37,39);(1,3)(2,6)(4,7)(5,8)(11,17)(15,21)(18,23)(22,24)(25,26)(27,29)(28,36)(30,33)(31,34)(32,38)(35,37)(39,40);(10,14)(15,21)(16,19)(22,24)(25,27,30,34)(26,31,33,29)(28,35,38,40)(32,37,36,39);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,39)(36,37)(38,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,30,38)(26,32,33,36)(27,35,34,40)(29,39,31,37);(1,4,5,2)(3,7,8,6)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29)(26,27)(28,37)(30,31)(32,40)(33,34)(35,36)(38,39)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9626475);