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 | #53493 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,19,23)(12,17,22,20)(13,21,24,18)(25,26,31,36)(27,34,39,32)(28,33,29,38)(30,35,40,37);(1,2)(3,4)(5,6)(7,8)(9,11,14,19)(10,15,16,23)(12,18,22,21)(13,20,24,17)(25,27)(26,32)(28,37)(29,35)(30,38)(31,39)(33,40)(34,36);(2,6)(3,7)(10,16)(11,19)(17,20)(18,21)(25,28,31,29)(26,33,36,38)(27,37,39,35)(30,32,40,34);(3,7)(4,8)(9,12,14,22)(10,17,16,20)(11,18,19,21)(13,23,24,15)(25,29,31,28)(26,33,36,38)(27,35,39,37)(30,32,40,34);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,21)(13,20)(14,19)(16,23)(17,24)(18,22)(25,27)(26,34)(28,35)(29,37)(30,38)(31,39)(32,36)(33,40);(1,4)(2,3)(5,8)(6,7)(9,13,14,24)(10,18,16,21)(11,20,19,17)(12,15,22,23)(25,30,31,40)(26,35,36,37)(27,38,39,33)(28,32,29,34)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8972853);