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 | #35693 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,16)(12,13)(14,18)(15,19)(17,24)(20,23)(21,22)(26,32)(27,37)(28,30)(35,40);(1,2,4,6)(3,5,7,8)(11,19)(15,16)(20,22)(21,23)(25,26)(27,38)(28,34)(29,40)(30,36)(31,32)(33,37)(35,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,19)(10,15,18,16)(12,22,17,20)(13,23,24,21)(25,27)(26,33)(28,29)(30,39)(31,37)(32,38)(34,35)(36,40);(9,12,14,17)(10,13,18,24)(11,20,19,22)(15,21,16,23)(25,28,31,30)(26,34,32,36)(27,29,37,39)(33,35,38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,22)(13,21)(14,19)(15,18)(17,20)(23,24)(25,29,31,39)(26,35,32,40)(27,28,37,30)(33,34,38,36);(1,3)(2,5)(4,7)(6,8)(9,13)(10,17)(11,21)(12,18)(14,24)(15,22)(16,20)(19,23)(25,30,31,28)(26,36,32,34)(27,39,37,29)(33,40,38,35)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,d^2=i,e^2=g,f^2=g,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, 8901248);