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 | #42063 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,21,24,16)(12,13,19,18)(14,22,23,20)(25,26,31,35)(27,29,33,40)(28,34,30,32)(36,39,37,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,24)(10,16,17,21)(12,22,19,20)(13,14,18,23)(25,27)(26,29)(28,36)(30,37)(31,33)(32,38)(34,39)(35,40);(2,5)(3,7)(10,17)(13,18)(14,23)(16,21)(25,28,31,30)(26,32,35,34)(27,36,33,37)(29,38,40,39);(3,7)(6,8)(9,12)(10,18)(11,22)(13,17)(14,21)(15,19)(16,23)(20,24);(1,3)(2,6)(4,7)(5,8)(9,13,15,18)(10,19,17,12)(11,23,24,14)(16,22,21,20)(25,29)(26,33)(27,35)(28,38)(30,39)(31,40)(32,37)(34,36);(1,2)(3,6)(4,5)(7,8)(9,14,15,23)(10,20,17,22)(11,18,24,13)(12,21,19,16)(25,30,31,28)(26,34,35,32)(27,37,33,36)(29,39,40,38)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8843474);