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 | |
4730 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)(11,17)(12,21)(13,22)(14,16)(15,20)(18,19)(23,24)(25,26)(27,28)(29,37)(30,31)(32,39)(33,36)(34,38)(35,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,20)(10,15,16,17)(12,18,22,24)(13,19,21,23)(25,27,30,33)(26,28,31,36)(29,34,39,40)(32,35,37,38);(1,3,5,6)(2,4,7,8)(10,16)(12,22)(15,17)(18,24)(26,31)(27,33)(32,37)(34,40);(9,12)(10,13)(11,18)(14,22)(15,19)(16,21)(17,23)(20,24)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,17)(12,18)(13,23)(14,20)(15,16)(19,21)(22,24)(25,28)(26,27)(29,38)(30,36)(31,33)(32,40)(34,37)(35,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,12)(11,19)(14,21)(15,18)(16,22)(17,24)(20,23)(25,29,30,39)(26,32,31,37)(27,34,33,40)(28,35,36,38)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,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, 9590533);