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 | |
4536 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)(4,7)(6,8)(9,10,14,16)(11,15,17,24)(12,20,22,13)(18,19,21,23)(25,26,31,34)(27,30,29,32)(28,33,38,40)(35,37,36,39);(1,2,6,8)(3,5,7,4)(9,11)(10,15)(12,18)(13,23)(14,17)(16,24)(19,20)(21,22)(27,29)(30,32)(35,36)(37,39);(1,3,6,7)(2,4,8,5)(11,17)(15,24)(18,21)(19,23)(25,27,31,29)(26,32,34,30)(28,36,38,35)(33,37,40,39);(1,4,6,5)(2,3,8,7)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,28)(26,33)(27,35)(29,36)(30,37)(31,38)(32,39)(34,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,13)(11,18)(14,22)(15,23)(16,20)(17,21)(19,24)(25,29,31,27)(26,30,34,32)(28,35,38,36)(33,39,40,37);(1,5,6,4)(2,7,8,3)(9,13)(10,12)(11,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,30)(26,27)(28,37)(29,34)(31,32)(33,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8733480);