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 | |
4511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,20)(12,22)(13,18)(14,23)(15,17)(16,21)(19,24)(26,31)(28,32)(33,39)(36,37);(1,3,5,8)(2,6,7,4)(9,11,15,21)(10,16,17,20)(12,14,18,24)(13,23,22,19)(25,26,30,31)(27,32,35,28)(29,33,38,39)(34,37,40,36);(1,2)(3,4)(5,7)(6,8)(10,17)(13,22)(16,20)(19,23)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,37,39,36);(1,4)(2,3)(5,6)(7,8)(9,12)(10,13)(11,14)(15,18)(16,23)(17,22)(19,20)(21,24)(25,28)(26,27)(29,37)(30,32)(31,35)(33,40)(34,39)(36,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,18)(11,19)(12,17)(14,16)(15,22)(20,24)(21,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,37)(36,39)(38,40);(1,4)(2,3)(5,6)(7,8)(9,14)(10,19)(11,18)(12,21)(13,16)(15,24)(17,23)(20,22)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,37)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8973124);