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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,17)(13,22)(15,18)(16,20)(19,21)(23,24)(25,26,31,35)(27,32,36,38)(28,33,37,40)(29,30,34,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,19)(13,23)(14,18)(15,20)(17,24)(21,22)(27,36)(28,37)(32,38)(33,40);(1,3,5,4)(2,6,7,8)(9,12,15,13)(10,17,18,22)(11,19,20,23)(14,21,16,24)(25,27)(26,32)(28,29)(30,33)(31,36)(34,37)(35,38)(39,40);(1,2)(3,6)(4,8)(5,7)(9,13,15,12)(10,17,18,22)(11,19,20,23)(14,24,16,21)(25,28)(26,33)(27,29)(30,32)(31,37)(34,36)(35,40)(38,39);(1,2)(3,6)(4,8)(5,7)(9,14,15,16)(10,11,18,20)(12,21,13,24)(17,19,22,23)(25,29)(26,30)(27,28)(31,34)(32,33)(35,39)(36,37)(38,40);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,19)(13,23)(15,20)(16,18)(17,21)(22,24)(25,30)(26,34)(27,33)(28,38)(29,35)(31,39)(32,37)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8599778);