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 | #2847 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10,15,18)(11,16,19,14)(12,13,17,23)(20,21,24,22)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(1,3,6,5)(2,7,8,4)(9,11,15,19)(10,16,18,14)(12,20,17,24)(13,21,23,22)(27,33)(30,37)(34,39)(38,40);(1,4,6,7)(2,5,8,3)(11,19)(14,16)(20,24)(21,22)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(2,8)(3,5)(9,12)(10,13)(11,20)(14,22)(15,17)(16,21)(18,23)(19,24)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40);(1,5,6,3)(2,4,8,7)(9,13)(10,17)(11,21)(12,18)(14,20)(15,23)(16,24)(19,22)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40);(1,4)(2,3)(5,8)(6,7)(9,14)(10,11)(12,22)(13,20)(15,16)(17,21)(18,19)(23,24)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8192702);