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 | #2541 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4302 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)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,26)(27,37)(28,39)(29,35)(30,31)(32,36)(33,40)(34,38);(1,2)(3,8)(4,5)(6,7)(10,11)(15,17)(16,18)(23,24)(25,27)(26,33)(28,29)(30,38)(31,39)(32,40)(34,35)(36,37);(1,3,6,5)(2,4,7,8)(25,28,32,38)(26,34,36,39)(27,29,40,30)(31,33,35,37);(1,4,6,8)(2,3,7,5)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,29,32,30)(26,31,36,35)(27,28,40,38)(33,39,37,34);(1,2)(3,4)(5,8)(6,7)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,30,32,29)(26,35,36,31)(27,38,40,28)(33,34,37,39);(1,5,6,3)(2,4,7,8)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,31,32,35)(26,30,36,29)(27,39,40,34)(28,33,38,37)Polycyclic presentation:
2^9|c^2=hi,d^2=gi,e^2=g,f^2=gi,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, 8608726);