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 | |
4247 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,16)(12,22)(13,17)(14,23)(15,19)(18,20)(21,24)(25,26,30,33)(27,29,32,35)(28,31,38,40)(34,37,36,39);(1,3)(2,6)(4,8)(5,7)(9,11,15,21)(10,16,19,24)(12,20,13,14)(17,23,22,18)(25,27)(26,29)(28,36)(30,32)(31,39)(33,35)(34,38)(37,40);(9,12,15,13)(10,17,19,22)(11,20,21,14)(16,23,24,18)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,40)(35,39);(1,2,5,8)(3,4,7,6)(9,13,15,12)(10,17,19,22)(11,14,21,20)(16,23,24,18)(25,26)(27,35)(28,31)(29,32)(30,33)(34,39)(36,37)(38,40);(1,3)(2,4)(5,7)(6,8)(9,10)(11,16)(12,17)(13,22)(14,18)(15,19)(20,23)(21,24)(25,29,30,35)(26,32,33,27)(28,37,38,39)(31,36,40,34);(1,4)(2,7)(3,8)(5,6)(9,14)(10,18)(11,12)(13,21)(15,20)(16,17)(19,23)(22,24)(25,28)(26,31)(27,36)(29,39)(30,38)(32,34)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10412187);