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 | |
4602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,18,16)(12,20,21,17)(19,24,22,23)(26,31)(27,33)(34,38)(35,40);(1,2)(3,7)(4,5)(6,8)(10,15)(11,18)(12,21)(23,24)(25,26,30,31)(27,32,33,28)(29,34,37,38)(35,39,40,36);(2,5)(6,7)(10,15)(13,16)(17,20)(23,24)(29,37)(34,38)(35,40)(36,39);(3,8)(6,7)(9,11,14,18)(10,16,15,13)(12,22,21,19)(17,23,20,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,19)(13,23)(14,21)(15,20)(16,24)(18,22)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(1,2)(3,6)(4,5)(7,8)(9,13,14,16)(10,18,15,11)(12,23,21,24)(17,22,20,19)(25,29,30,37)(26,34,31,38)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8269281);