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 | |
4542 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)(3,4)(5,7)(6,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,32,31)(27,29,33,28)(30,38,39,34)(35,36,40,37);(1,3)(2,6)(4,7)(5,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,29)(28,31)(30,35)(32,33)(34,37)(36,38)(39,40);(2,7)(4,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,32,29)(26,33,31,27)(30,36,39,37)(34,40,38,35);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,29,32,28)(26,33,31,27)(30,37,39,36)(34,40,38,35);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,30,32,39)(26,34,31,38)(27,35,33,40)(28,36,29,37);(1,4,5,6)(2,3,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,31,32,26)(27,28,33,29)(30,38,39,34)(35,36,40,37)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8092080);