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 | |
4609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,13)(12,20)(14,17)(15,19)(16,21)(18,24)(22,23)(25,26,31,36)(27,37,39,32)(28,30,35,33)(29,40,38,34);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,24)(20,22)(21,23)(25,27)(26,32)(28,29)(30,34)(31,39)(33,40)(35,38)(36,37);(1,2)(3,4)(5,7)(6,8)(12,21)(16,20)(18,23)(22,24)(25,28,31,35)(26,33,36,30)(27,29,39,38)(32,40,37,34);(1,3,5,8)(2,6,7,4)(9,12)(10,16)(11,18)(13,22)(14,21)(15,24)(17,20)(19,23)(25,28)(26,33)(27,38)(29,39)(30,36)(31,35)(32,34)(37,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,15)(10,11,17,19)(12,22,21,24)(16,18,20,23)(25,29,31,38)(26,34,36,40)(27,28,39,35)(30,37,33,32);(1,4,5,6)(2,3,7,8)(9,11)(10,13)(12,23)(14,19)(15,17)(16,24)(18,21)(20,22)(25,30,31,33)(26,35,36,28)(27,34,39,40)(29,32,38,37)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8626126);