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 | |
4576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,12)(11,15,17,19)(13,16,22,20)(18,23,21,24)(25,26,31,34)(27,32,38,33)(28,39,29,30)(35,37,36,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,21)(14,17)(16,24)(18,22)(20,23)(25,27,31,38)(26,32,34,33)(28,36,29,35)(30,37,39,40);(2,4)(6,7)(11,17)(15,19)(18,21)(23,24)(25,28,31,29)(26,30,34,39)(27,35,38,36)(32,40,33,37);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,15,17,19)(13,20,22,16)(18,23,21,24)(25,29,31,28)(26,30,34,39)(27,36,38,35)(32,40,33,37);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,24)(21,22)(25,30)(26,28)(27,37)(29,34)(31,39)(32,36)(33,35)(38,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,27)(26,33)(28,35)(29,36)(30,37)(31,38)(32,34)(39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10259632);