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 | |
4626 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)(3,5)(4,8)(6,7)(9,10)(11,19)(12,23)(13,18)(14,15)(16,22)(17,21)(20,24)(25,26,30,32)(27,33,34,29)(28,31,36,39)(35,40,37,38);(2,7)(3,8)(9,11,15,22)(10,16,14,19)(12,20,21,13)(17,24,23,18)(25,26)(27,29)(28,31)(30,32)(33,34)(35,38)(36,39)(37,40);(1,3,6,8)(2,4,7,5)(10,14)(12,21)(13,20)(16,19)(27,34)(28,36)(29,33)(31,39);(1,4,6,5)(2,3,7,8)(9,12)(10,17)(11,20)(13,22)(14,23)(15,21)(16,24)(18,19)(25,27)(26,29)(28,37)(30,34)(31,40)(32,33)(35,36)(38,39);(1,2)(3,4)(5,8)(6,7)(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,23,19,17)(25,28)(26,31)(27,35)(29,38)(30,36)(32,39)(33,40)(34,37);(1,5,6,4)(2,8,7,3)(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,24,20,18)(25,29)(26,27)(28,38)(30,33)(31,35)(32,34)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=i,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8924141);