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 | |
4592 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)(9,10,14,16)(11,13,19,15)(12,20,21,17)(18,24,22,23)(25,26,30,32)(27,31,35,40)(28,33,37,39)(29,34,36,38);(3,7)(5,8)(9,11,14,19)(10,15,16,13)(12,18,21,22)(17,24,20,23)(25,27,30,35)(26,31,32,40)(28,36,37,29)(33,38,39,34);(1,2)(3,5)(4,6)(7,8)(10,16)(12,21)(13,15)(18,22)(26,32)(27,35)(28,37)(34,38);(9,12,14,21)(10,17,16,20)(11,18,19,22)(13,23,15,24)(25,28)(26,33)(27,36)(29,35)(30,37)(31,38)(32,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,22)(14,19)(15,16)(17,24)(18,21)(20,23)(25,29,30,36)(26,34,32,38)(27,37,35,28)(31,39,40,33);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,23)(14,15)(16,19)(17,18)(20,22)(21,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,40)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8824957);