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 | #2410 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(26,31)(27,33)(28,35)(38,40);(3,7)(6,8)(9,10,14,15)(11,16,20,22)(12,19,21,17)(13,18,23,24)(25,26,30,31)(27,29,33,39)(28,36,35,32)(34,40,37,38);(1,2)(3,6)(4,5)(7,8)(10,15)(12,21)(16,22)(18,24)(25,27,30,33)(26,29,31,39)(28,37,35,34)(32,40,36,38);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,18)(13,17)(14,20)(15,22)(19,23)(21,24)(25,26)(27,29)(28,32)(30,31)(33,39)(34,38)(35,36)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,21)(10,17,15,19)(11,18,20,24)(13,22,23,16)(25,28,30,35)(26,32,31,36)(27,34,33,37)(29,38,39,40);(9,13)(10,18)(11,19)(12,22)(14,23)(15,24)(16,21)(17,20)(25,29)(26,33)(27,31)(28,38)(30,39)(32,37)(34,36)(35,40)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8525366);