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 | #49555 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4207 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)(11,19)(12,15)(13,17)(14,22)(16,20)(18,23)(21,24)(25,26)(27,34)(28,32)(29,39)(30,35)(31,36)(33,37)(38,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,20)(10,16,12,19)(13,23,24,14)(17,22,21,18)(25,27)(26,30)(28,37)(29,38)(31,35)(32,39)(33,40)(34,36);(1,3,5,6)(2,4,7,8)(10,12)(16,19)(17,21)(18,22)(25,28,31,38)(26,32,36,40)(27,37,35,29)(30,39,34,33);(9,12,15,10)(11,19,20,16)(13,21,24,17)(14,22,23,18)(25,29,31,37)(26,33,36,39)(27,38,35,28)(30,40,34,32);(1,2)(3,4)(5,7)(6,8)(9,13,15,24)(10,17,12,21)(11,14,20,23)(16,18,19,22)(25,27)(26,34)(28,37)(29,38)(30,36)(31,35)(32,33)(39,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,18)(11,13)(12,22)(15,23)(16,17)(19,21)(20,24)(25,30,31,34)(26,35,36,27)(28,39,38,33)(29,40,37,32)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=g,e^2=h,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, 9591361);