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 | |
4281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,19,20,14)(12,21,24,17)(13,23,22,18)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(1,2,5,4)(3,6,7,8)(9,11,15,20)(10,14,16,19)(12,22,24,13)(17,18,21,23)(28,36)(32,38)(35,39)(37,40);(2,4)(6,8)(10,16)(14,19)(17,21)(18,23)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(3,7)(6,8)(9,12,15,24)(10,17,16,21)(11,13,20,22)(14,23,19,18)(25,28,31,36)(26,32,30,38)(27,35,33,39)(29,37,34,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,12)(14,21)(15,22)(16,23)(17,19)(20,24)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,4,5,2)(3,8,7,6)(9,14,15,19)(10,11,16,20)(12,23,24,18)(13,21,22,17)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8233187);