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 | |
4323 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)(3,5)(4,7)(6,8)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,24)(21,23)(25,26)(27,32)(28,29)(30,40)(31,34)(33,38)(35,39)(36,37);(2,3)(6,7)(9,11)(10,16)(12,14)(13,23)(15,20)(17,21)(18,22)(19,24)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(1,2,5,3)(4,6,8,7)(11,20)(14,21)(16,24)(18,23)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(1,3,5,2)(4,7,8,6)(9,12)(10,13)(11,21)(14,20)(15,17)(16,18)(19,22)(23,24)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(1,4,5,8)(2,6,3,7)(9,13)(10,17)(11,18)(12,19)(14,16)(15,22)(20,23)(21,24)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36);(9,14,15,21)(10,18,19,23)(11,12,20,17)(13,16,22,24)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325338);