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 | |
4429 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)(3,7)(9,10,15,14)(11,19,22,16)(12,17,21,23)(13,24,20,18)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2,4,5)(3,6,7,8)(9,11,15,22)(10,16,14,19)(12,20,21,13)(17,24,23,18)(28,36)(31,38)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(10,14)(16,19)(17,23)(18,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,20)(13,22)(14,23)(15,21)(16,24)(18,19)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(1,3,4,7)(2,6,5,8)(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,23,19,17)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,24,20,18)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325604);