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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4071 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)(11,15)(12,16)(13,24)(14,18)(17,21)(19,20)(22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2,4,6)(3,7,8,5)(9,10,14,18)(11,19,20,15)(12,22,23,16)(13,24,21,17)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(3,8)(5,7)(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,13)(14,23)(15,24)(17,19)(18,22)(20,21);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,12)(14,21)(15,22)(16,19)(18,24)(20,23)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=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, 9431012);