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 | #42541 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4379 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,6,7,8)(10,15)(11,18)(12,20)(21,23)(25,26)(27,38)(28,31)(29,36)(30,35)(32,37)(33,40)(34,39);(2,5)(6,8)(9,10,14,15)(11,13,18,16)(12,17,20,22)(19,21,24,23)(25,27)(26,33)(28,29)(30,39)(31,35)(32,40)(34,36)(37,38);(3,7)(6,8)(9,11,14,18)(10,16,15,13)(12,19,20,24)(17,23,22,21)(25,28,32,39)(26,34,37,31)(27,29,40,30)(33,36,38,35);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,29,32,30)(26,35,37,36)(27,28,40,39)(31,38,34,33);(9,12,14,20)(10,17,15,22)(11,19,18,24)(13,21,16,23)(25,30,32,29)(26,36,37,35)(27,39,40,28)(31,33,34,38);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,18,15,11)(12,21,20,23)(17,24,22,19)(25,31,32,34)(26,28,37,39)(27,35,40,36)(29,38,30,33)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=hi,e^2=h,f^2=g,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, 8095757);