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 | #41385 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(25,26)(27,35)(28,37)(29,34)(30,38)(31,32)(33,40)(36,39);(2,3)(6,7)(9,10,15,17)(11,18,12,13)(14,22,23,16)(19,21,20,24)(26,32)(29,39)(30,35)(33,37);(1,3,5,2)(4,7,8,6)(9,11,15,12)(10,13,17,18)(14,19,23,20)(16,21,22,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,35)(36,40)(37,39);(1,4)(2,6)(3,7)(5,8)(9,12,15,11)(10,13,17,18)(14,20,23,19)(16,21,22,24)(25,28,31,40)(26,33,32,37)(27,36,38,34)(29,30,39,35);(9,13)(10,11)(12,17)(14,21)(15,18)(16,19)(20,22)(23,24)(25,29,31,39)(26,34,32,36)(27,37,38,33)(28,30,40,35);(9,14)(10,16)(11,19)(12,20)(13,21)(15,23)(17,22)(18,24)(25,30)(26,27)(28,39)(29,40)(31,35)(32,38)(33,36)(34,37)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=i,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9174181);