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 | |
4707 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)(3,7)(4,5)(6,8)(10,15)(11,17)(18,22)(19,24)(25,26)(27,34)(28,36)(29,33)(30,31)(32,37)(35,39)(38,40);(3,8)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,27,31,33)(26,29,30,34)(28,35,37,40)(32,39,36,38);(1,3,5,8)(2,6,4,7)(11,17)(12,16)(13,21)(18,22)(26,30)(28,37)(29,34)(35,40);(3,8)(6,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,28,31,37)(26,32,30,36)(27,35,33,40)(29,39,34,38);(1,3)(2,7)(4,6)(5,8)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29,31,34)(26,33,30,27)(28,38,37,39)(32,35,36,40);(1,4,5,2)(3,6,8,7)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,30,31,26)(27,34,33,29)(28,36,37,32)(35,38,40,39)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8902972);