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 | |
4401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(16,23)(19,24)(25,26,30,35)(27,31,37,40)(28,33,38,32)(29,36,39,34);(1,2)(3,8)(4,6)(5,7)(9,10,14,15)(11,18,17,13)(12,16,21,23)(19,20,24,22)(25,27)(26,31)(28,34)(29,33)(30,37)(32,39)(35,40)(36,38);(1,3,5,6)(2,4,7,8)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(1,4,5,8)(2,3,7,6)(9,11,14,17)(10,13,15,18)(12,19,21,24)(16,22,23,20)(25,28)(26,33)(27,34)(29,31)(30,38)(32,35)(36,37)(39,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,24)(18,20)(25,29)(26,34)(27,33)(28,31)(30,39)(32,37)(35,36)(38,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,20)(14,18)(15,17)(16,24)(19,23)(21,22)(25,27)(26,31)(28,36)(29,32)(30,37)(33,39)(34,38)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=hi,d^2=ghi,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, 8604697);