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 | |
4508 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,22)(16,18)(17,24)(19,20)(21,23)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(2,3)(6,7)(9,11,16,15)(10,13,18,12)(14,19,24,23)(17,21,22,20)(28,36)(33,38)(35,37)(39,40);(4,8)(6,7)(9,12,16,13)(10,11,18,15)(14,20,24,21)(17,19,22,23)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,3,5,2)(4,7,8,6)(9,13,16,12)(10,11,18,15)(14,21,24,20)(17,19,22,23)(25,28,31,36)(26,33,30,38)(27,35,29,37)(32,39,34,40);(9,14,16,24)(10,17,18,22)(11,19,15,23)(12,20,13,21)(25,29,31,27)(26,34,30,32)(28,37,36,35)(33,40,38,39);(1,4)(2,6)(3,7)(5,8)(9,15,16,11)(10,12,18,13)(14,23,24,19)(17,20,22,21)(25,30,31,26)(27,34,29,32)(28,38,36,33)(35,40,37,39)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8092026);