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 | |
4552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,21)(14,15)(16,18)(17,22)(19,23)(20,24)(25,26)(27,36)(28,32)(29,34)(30,35)(31,39)(33,38)(37,40);(1,2)(3,6)(4,7)(5,8)(10,15)(12,16)(17,21)(20,23)(25,27)(26,31)(28,37)(29,38)(30,39)(32,34)(33,40)(35,36);(1,3)(2,5)(4,8)(6,7)(10,15)(11,18)(13,22)(20,23)(25,28,30,38)(26,32,35,33)(27,37,39,29)(31,34,36,40);(3,8)(5,6)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,28)(26,33)(27,37)(29,39)(30,38)(31,40)(32,35)(34,36);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29,30,37)(26,34,35,40)(27,38,39,28)(31,32,36,33);(1,2)(3,6)(4,7)(5,8)(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,26)(27,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40)Polycyclic presentation:
2^9|c^2=gh,d^2=h,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,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, 8811708);