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 | |
4507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26)(27,30)(28,33)(29,34)(31,32)(35,40)(36,38)(37,39);(1,2)(3,5)(4,7)(6,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,27,31,35)(26,30,32,40)(28,36,37,29)(33,38,39,34);(1,3)(2,5)(4,8)(6,7)(12,21)(16,20)(19,22)(23,24)(26,32)(27,35)(28,37)(34,38);(1,3,4,8)(2,5,7,6)(9,12,14,21)(10,16,17,20)(11,19,13,22)(15,23,18,24)(25,28)(26,33)(27,36)(29,35)(30,38)(31,37)(32,39)(34,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,15,17,18)(12,19,21,22)(16,24,20,23)(25,29,31,36)(26,34,32,38)(27,37,35,28)(30,39,40,33);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,21)(22,24)(25,30)(26,35)(27,32)(28,38)(29,33)(31,40)(34,37)(36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9034761);