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 | |
4563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,16)(11,18,13,15)(12,20,22,17)(19,23,21,24)(26,31)(27,33)(32,35)(34,39);(1,3,6,8)(2,7,4,5)(9,11,14,13)(10,15,16,18)(12,21,22,19)(17,24,20,23)(25,26,30,31)(27,29,33,38)(28,35,36,32)(34,40,39,37);(2,4)(3,8)(10,16)(15,18)(17,20)(23,24)(28,36)(32,35)(34,39)(37,40);(3,8)(5,7)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,40)(36,39);(1,4,6,2)(3,7,8,5)(9,10)(11,18)(12,17)(13,15)(14,16)(19,24)(20,22)(21,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,35)(33,39)(38,40);(1,5,6,7)(2,8,4,3)(9,13,14,11)(10,15,16,18)(12,19,22,21)(17,24,20,23)(25,29)(26,33)(27,31)(28,37)(30,38)(32,39)(34,35)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,e^2=i,f^2=h,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, 8529129);