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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4132 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,18,13)(12,16,20,23)(19,21,24,22)(26,30)(27,33)(28,36)(37,40);(3,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,21)(20,24)(22,23)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(1,3,5,8)(2,6,4,7)(11,18)(12,20)(13,15)(16,23)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(3,8)(6,7)(9,12)(10,16)(11,19)(13,22)(14,20)(15,21)(17,23)(18,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,36)(33,39)(34,40);(1,3)(2,7)(4,6)(5,8)(9,13,14,15)(10,11,17,18)(12,21,20,22)(16,24,23,19)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,4,5,2)(3,6,8,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,21)(20,24)(22,23)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8909594);