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 |
| prime radical | #35438 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4537 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)(3,5)(4,7)(6,8)(9,10,15,17)(11,18,12,16)(13,19,23,22)(14,21,24,20)(25,26)(27,29)(28,33)(30,39)(31,32)(34,38)(35,36)(37,40);(1,2,6,8)(3,5,7,4)(9,11,15,12)(10,16,17,18)(13,20,23,21)(14,19,24,22)(25,27,31,30)(26,29,32,39)(28,34,37,36)(33,38,40,35);(1,3,6,7)(2,4,8,5)(10,17)(14,24)(16,18)(19,22)(26,32)(27,30)(33,40)(34,36);(1,4,6,5)(2,3,8,7)(9,12,15,11)(10,18,17,16)(13,21,23,20)(14,22,24,19)(25,28)(26,33)(27,34)(29,38)(30,36)(31,37)(32,40)(35,39);(1,2)(3,4)(5,7)(6,8)(9,13)(10,19)(11,21)(12,20)(14,16)(15,23)(17,22)(18,24)(25,29)(26,27)(28,35)(30,32)(31,39)(33,36)(34,40)(37,38);(1,5,6,4)(2,7,8,3)(9,14,15,24)(10,20,17,21)(11,19,12,22)(13,16,23,18)(25,30,31,27)(26,29,32,39)(28,36,37,34)(33,38,40,35)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806107);