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 | |
4398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,18)(12,22)(13,17)(14,15)(16,21)(19,20)(23,24)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,37,40,38);(1,3,4,8)(2,5,6,7)(10,15)(12,17)(16,18)(19,23)(28,36)(32,39)(35,40)(37,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,16,15,18)(12,19,17,23)(13,20,22,24)(25,27,30,33)(26,31,34,29)(28,35,36,40)(32,38,39,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,19)(14,17)(15,22)(16,20)(18,24)(21,23)(25,28,30,36)(26,32,34,39)(27,35,33,40)(29,37,31,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,23)(22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,13)(10,17)(11,20)(12,15)(14,22)(16,23)(18,19)(21,24)(25,29,30,31)(26,33,34,27)(28,37,36,38)(32,40,39,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=hi,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440668);