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 | |
4420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,20)(11,19,14,15)(12,18,13,17)(21,22,23,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,16,14)(10,15,20,19)(12,21,13,23)(17,22,18,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(2,5)(6,8)(9,12,16,13)(10,17,20,18)(11,21,14,23)(15,22,19,24)(25,27,29,35)(26,31,33,30)(28,34,37,40)(32,39,36,38);(3,7)(6,8)(9,13,16,12)(10,18,20,17)(11,21,14,23)(15,22,19,24)(25,26)(27,31)(28,36)(29,33)(30,35)(32,37)(34,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,14,16,11)(10,19,20,15)(12,21,13,23)(17,22,18,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,3,4,7)(2,6,5,8)(9,15,16,19)(10,11,20,14)(12,22,13,24)(17,21,18,23)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8450928);