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 | #38132 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4393 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)(3,4)(5,7)(6,8)(9,10,15,19)(11,16,20,22)(12,17,21,24)(13,14,18,23)(25,26,32,36)(27,34,28,33)(29,31,30,35)(37,40,38,39);(1,3)(2,6)(4,7)(5,8)(11,20)(12,21)(16,22)(17,24)(25,27,32,28)(26,33,36,34)(29,37,30,38)(31,39,35,40);(9,11,15,20)(10,16,19,22)(12,14,21,23)(13,17,18,24)(25,28,32,27)(26,34,36,33)(29,38,30,37)(31,40,35,39);(2,7)(3,8)(9,12)(10,17)(11,14)(13,22)(15,21)(16,18)(19,24)(20,23)(25,29,32,30)(26,35,36,31)(27,37,28,38)(33,40,34,39);(1,4,5,6)(2,3,7,8)(9,13)(10,14)(11,17)(12,22)(15,18)(16,21)(19,23)(20,24)(25,30,32,29)(26,31,36,35)(27,38,28,37)(33,39,34,40);(1,3)(2,4)(5,8)(6,7)(9,14)(10,18)(11,21)(12,20)(13,19)(15,23)(16,24)(17,22)(25,31)(26,30)(27,39)(28,40)(29,36)(32,35)(33,38)(34,37)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=hi,d^2=h,e^2=g,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, 8603114);