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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,17)(12,15)(13,16)(14,20)(18,22)(19,23)(21,24)(25,26,31,35)(27,32,37,40)(28,29,33,38)(30,39,36,34);(1,3,5,8)(2,6,7,4)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27,31,37)(26,32,35,40)(28,36,33,30)(29,34,38,39);(1,3)(2,4)(5,8)(6,7)(10,12)(16,17)(18,20)(23,24)(25,28,31,33)(26,29,35,38)(27,30,37,36)(32,39,40,34);(1,4)(2,3)(5,6)(7,8)(9,12,15,10)(11,16,13,17)(14,20,22,18)(19,23,21,24)(25,28)(26,29)(27,36)(30,37)(31,33)(32,34)(35,38)(39,40);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29)(26,33)(27,34)(28,35)(30,32)(31,38)(36,40)(37,39);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30,31,36)(26,34,35,39)(27,28,37,33)(29,32,38,40)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081609);