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 | #47446 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,13,17,15)(12,16,22,20)(19,23,24,21)(25,26,31,36)(27,32,29,34)(28,38,37,33)(30,35,39,40);(1,2)(3,6)(4,7)(5,8)(9,11,14,17)(10,15,18,13)(12,19,22,24)(16,21,20,23)(25,27,31,29)(26,32,36,34)(28,39,37,30)(33,35,38,40);(1,3)(2,5)(4,8)(6,7)(25,28,31,37)(26,33,36,38)(27,30,29,39)(32,40,34,35);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,20)(13,17)(14,18)(16,22)(19,23)(21,24)(25,29,31,27)(26,34,36,32)(28,30,37,39)(33,40,38,35);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,23)(14,22)(15,21)(17,24)(18,20)(25,30)(26,35)(27,37)(28,29)(31,39)(32,33)(34,38)(36,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,19,20,24)(25,26)(27,32)(28,33)(29,34)(30,40)(31,36)(35,39)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,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^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9994563);