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 | |
4624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,19,20,16)(12,17,23,22)(13,18,24,21)(25,26,31,32)(27,29,28,33)(30,34,38,39)(35,37,36,40);(3,7)(5,8)(10,15)(11,20)(17,22)(18,21)(26,32)(27,28)(34,39)(35,36);(1,2)(3,5)(4,6)(7,8)(25,27,31,28)(26,33,32,29)(30,35,38,36)(34,40,39,37);(1,3)(2,5)(4,7)(6,8)(9,11,14,20)(10,16,15,19)(12,21,23,18)(13,17,24,22)(25,28,31,27)(26,29,32,33)(30,36,38,35)(34,37,39,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,29)(26,27)(28,32)(30,37)(31,33)(34,35)(36,39)(38,40);(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,30,31,38)(26,34,32,39)(27,35,28,36)(29,37,33,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=i,d^2=g,f^2=g,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, 8602592);