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 | |
4556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10,13,16)(11,14,18,17)(12,15,21,20)(19,22,24,23);(1,3)(2,7)(4,5)(6,8)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,21)(22,24)(25,26)(27,32)(28,29)(30,31)(33,37)(34,38)(35,40)(36,39);(1,4,6,7)(2,3,5,8)(11,18)(14,17)(19,24)(22,23)(26,31)(28,37)(32,39)(34,40);(2,5)(3,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,18)(19,23)(22,24)(25,27)(26,32)(28,34)(29,38)(30,36)(31,39)(33,35)(37,40);(1,4)(2,3)(5,8)(6,7)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,28)(26,33)(27,34)(29,31)(30,37)(32,35)(36,40)(38,39);(1,5,6,2)(3,4,8,7)(9,12)(10,15)(11,19)(13,21)(14,22)(16,20)(17,23)(18,24)(25,29,30,33)(26,28,31,37)(27,35,36,38)(32,40,39,34)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541077);