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 | #45550 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,15,17)(11,19,13,16)(12,22,23,18)(14,20,24,21)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,17,19)(12,20,23,21)(14,18,24,22)(26,27)(31,33)(32,34)(38,39);(1,2)(3,6)(4,5)(7,8)(10,17)(14,24)(16,19)(18,22)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,19,24,16)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,19,17,16)(12,21,23,20)(14,22,24,18)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39);(9,14)(10,20)(11,18)(12,19)(13,22)(15,24)(16,23)(17,21)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=gi,e^2=g,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, 8198833);