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 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,14)(11,19,21,16)(12,17,23,22)(13,24,20,18)(25,26)(27,31)(28,32)(29,37)(30,34)(33,38)(35,36)(39,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,20)(13,23)(14,19)(15,21)(17,24)(18,22)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,23)(10,17,14,22)(11,20,21,13)(16,24,19,18)(25,27,30,32)(26,28,34,31)(29,35,38,40)(33,36,37,39);(2,6)(5,7)(9,13)(10,18)(11,12)(14,24)(15,20)(16,17)(19,22)(21,23)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,2)(3,5)(4,6)(7,8)(9,11)(10,19)(12,20)(13,23)(14,16)(15,21)(17,18)(22,24)(25,26)(27,28)(29,37)(30,34)(31,32)(33,38)(35,39)(36,40);(1,3)(2,7)(4,8)(5,6)(9,14,15,10)(11,19,21,16)(12,22,23,17)(13,24,20,18)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10125697);