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 | #42471 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,13,18,23)(12,17,21,14)(19,22,24,20)(25,26)(27,38)(28,36)(29,31)(30,34)(32,37)(33,40)(35,39);(1,2)(3,4)(5,6)(7,8)(9,11)(10,13)(12,19)(14,20)(15,18)(16,23)(17,22)(21,24)(25,27,32,40)(26,33,37,38)(28,39,30,29)(31,34,35,36);(2,6)(4,7)(10,16)(11,18)(14,17)(19,24)(25,28,32,30)(26,34,37,36)(27,39,40,29)(31,33,35,38);(2,6)(3,8)(9,12)(10,17)(11,19)(13,22)(14,16)(15,21)(18,24)(20,23)(25,29)(26,35)(27,28)(30,40)(31,37)(32,39)(33,36)(34,38);(1,3)(2,7)(4,6)(5,8)(9,13,15,23)(10,11,16,18)(12,20,21,22)(14,19,17,24)(25,30,32,28)(26,36,37,34)(27,39,40,29)(31,33,35,38);(1,4,5,7)(2,3,6,8)(9,14,15,17)(10,12,16,21)(11,20,18,22)(13,19,23,24)(25,31,32,35)(26,29,37,39)(27,34,40,36)(28,38,30,33)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306176);