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 | #2617 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4061 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)(9,10,14,17)(11,12,15,20)(13,16,22,24)(18,19,21,23)(25,26,32,30)(27,28,37,33)(29,31,39,34)(35,36,40,38);(1,2,4,6)(3,5,8,7)(9,11,14,15)(10,12,17,20)(13,21,22,18)(16,23,24,19)(25,27,32,37)(26,33,30,28)(29,35,39,40)(31,38,34,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,18)(14,15)(16,19)(17,20)(21,22)(23,24)(25,28)(26,27)(29,36)(30,37)(31,35)(32,33)(34,40)(38,39);(1,3,4,8)(2,5,6,7)(25,29,32,39)(26,34,30,31)(27,35,37,40)(28,38,33,36);(9,12)(10,15)(11,17)(13,19)(14,20)(16,21)(18,24)(22,23)(25,30,32,26)(27,33,37,28)(29,31,39,34)(35,36,40,38);(9,13)(10,16)(11,18)(12,19)(14,22)(15,21)(17,24)(20,23)(25,31)(26,29)(27,36)(28,35)(30,39)(32,34)(33,40)(37,38)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150855);