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 | |
4474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,7)(9,10,14,17)(11,15,20,13)(12,16,22,24)(18,23,19,21)(25,26,30,34)(27,29,37,33)(28,32,39,31)(35,40,36,38);(1,2,4,5)(3,6,8,7)(9,11,14,20)(10,15,17,13)(12,19,22,18)(16,21,24,23)(25,27)(26,29)(28,36)(30,37)(31,40)(32,38)(33,34)(35,39);(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20)(25,28)(26,31)(27,35)(29,38)(30,39)(32,34)(33,40)(36,37);(1,2,4,5)(3,7,8,6)(9,12)(10,16)(11,19)(13,23)(14,22)(15,21)(17,24)(18,20)(25,28)(26,32)(27,36)(29,38)(30,39)(31,34)(33,40)(35,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,18)(14,20)(15,17)(16,21)(19,22)(23,24)(25,29,30,33)(26,27,34,37)(28,38,39,40)(31,35,32,36);(1,3)(2,6)(4,8)(5,7)(9,13)(10,11)(12,21)(14,15)(16,18)(17,20)(19,24)(22,23)(25,27)(26,33)(28,35)(29,34)(30,37)(31,40)(32,38)(36,39)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10327519);