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 | #45476 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4343 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)(3,7)(4,5)(6,8)(12,20)(15,22)(18,24)(21,23)(25,26)(27,37)(28,29)(30,34)(31,39)(32,36)(33,40)(35,38);(3,8)(6,7)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(25,27)(26,33)(28,38)(29,39)(30,31)(32,40)(34,35)(36,37);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,32,30)(26,34,36,29)(27,38,40,31)(33,35,37,39);(1,2,5,4)(3,7,8,6)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,29,32,34)(26,28,36,30)(27,39,40,35)(31,33,38,37);(1,3)(2,7)(4,6)(5,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,30,32,28)(26,34,36,29)(27,38,40,31)(33,39,37,35);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,31,32,38)(26,35,36,39)(27,30,40,28)(29,33,34,37)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,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, 9624754);