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 | |
4417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,14)(11,15,17,18)(12,20,22,16)(19,24,21,23)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,2)(3,7)(4,6)(5,8)(10,14)(11,17)(16,20)(19,21)(26,31)(27,36)(33,38)(34,37);(1,3)(2,5)(4,8)(6,7)(10,14)(15,18)(16,20)(23,24);(2,6)(5,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,22)(21,23)(25,27)(26,32)(28,34)(29,38)(30,36)(31,35)(33,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,17)(10,15,14,18)(12,21,22,19)(16,24,20,23)(25,28,30,39)(26,33,31,38)(27,37,36,34)(29,35,40,32);(1,3)(2,7)(4,8)(5,6)(9,12,13,22)(10,16,14,20)(11,19,17,21)(15,23,18,24)(25,29)(26,34)(27,38)(28,32)(30,40)(31,37)(33,36)(35,39)Polycyclic presentation:
2^9|a^2=h,e^2=g,f^2=h,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, 10126993);