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 | |
4322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,26)(27,28)(29,37)(30,32)(31,39)(33,35)(34,38)(36,40);(1,2)(3,8)(4,6)(5,7)(10,14)(15,17)(16,20)(22,23)(25,27,30,35)(26,28,32,33)(29,34,39,40)(31,36,37,38);(1,3,5,4)(2,6,7,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,26,30,32)(27,33,35,28)(29,31,39,37)(34,38,40,36);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40);(1,4,5,3)(2,6,7,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,28)(26,27)(29,38)(30,33)(31,40)(32,35)(34,37)(36,39);(1,2)(3,6)(4,8)(5,7)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29)(26,31)(27,34)(28,36)(30,39)(32,37)(33,38)(35,40)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9613980);