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 | |
4626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,12,19,15)(13,21,23,17)(18,24,22,20)(25,26)(27,30)(28,33)(29,34)(31,32)(35,39)(36,38)(37,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,19)(10,15,16,12)(13,18,23,22)(17,24,21,20)(25,27)(26,30)(28,29)(31,35)(32,39)(33,34)(36,37)(38,40);(1,3,5,6)(2,4,7,8)(10,16)(12,15)(17,21)(20,24)(26,32)(27,35)(28,36)(34,40);(9,12)(10,11)(13,20)(14,15)(16,19)(17,18)(21,22)(23,24)(25,28,31,36)(26,33,32,38)(27,29,35,37)(30,34,39,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,22)(14,19)(15,16)(17,24)(18,23)(20,21)(25,29)(26,34)(27,36)(28,35)(30,38)(31,37)(32,40)(33,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,18)(12,20)(14,23)(15,24)(16,21)(19,22)(25,30,31,39)(26,35,32,27)(28,34,36,40)(29,33,37,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=gh,f^2=gh,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, 9590854);