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 | #37744 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4558 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)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(25,26)(27,32)(28,29)(30,36)(31,35)(33,39)(34,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,37)(26,32,35,40)(28,38,39,30)(29,34,33,36);(2,5)(3,7)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22)(25,28,31,39)(26,33,35,29)(27,30,37,38)(32,34,40,36);(3,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,26)(27,32)(28,33)(29,39)(30,34)(31,35)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,29,31,33)(26,28,35,39)(27,36,37,34)(30,40,38,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,30)(26,34)(27,28)(29,40)(31,38)(32,33)(35,36)(37,39)Polycyclic presentation:
2^9|b^2=gh,c^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8842826);