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 | #36744 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,13,17,15)(12,16,21,23)(19,22,24,20)(26,31)(27,34)(28,36)(37,39);(1,2)(3,8)(4,6)(5,7)(9,11,14,17)(10,15,18,13)(12,19,21,24)(16,20,23,22)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(1,3,5,6)(2,4,7,8)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,4,5,8)(2,3,7,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,20)(21,24)(22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,20)(17,24)(18,23)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(1,4)(2,6)(3,7)(5,8)(9,13,14,15)(10,17,18,11)(12,20,21,22)(16,19,23,24)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,d^2=hi,e^2=h,f^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, 8604819);