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 | #59180 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4203 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)(3,5)(4,7)(6,8)(9,10,15,18)(11,14,17,23)(12,13,16,22)(19,20,21,24)(25,26,32,29)(27,33,30,39)(28,37,36,34)(31,35,38,40);(2,3)(6,7)(9,11)(10,14)(12,19)(13,20)(15,17)(16,21)(18,23)(22,24)(25,27,32,30)(26,33,29,39)(28,31,36,38)(34,40,37,35);(1,2,5,3)(4,6,8,7)(11,17)(14,23)(19,21)(20,24)(25,28)(26,34)(27,31)(29,37)(30,38)(32,36)(33,40)(35,39);(1,3,5,2)(4,7,8,6)(9,12)(10,13)(11,19)(14,20)(15,16)(17,21)(18,22)(23,24)(25,29,32,26)(27,33,30,39)(28,37,36,34)(31,40,38,35);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,20)(12,18)(14,21)(15,22)(17,24)(19,23)(25,30,32,27)(26,33,29,39)(28,38,36,31)(34,40,37,35);(9,14,15,23)(10,17,18,11)(12,20,16,24)(13,21,22,19)(25,31,32,38)(26,35,29,40)(27,36,30,28)(33,34,39,37)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231828);