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 | |
4630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(25,26)(27,33)(28,29)(30,39)(31,32)(34,38)(35,36)(37,40);(2,5)(6,8)(10,11)(15,17)(16,18)(22,23)(26,32)(29,34)(33,40)(35,39);(1,2)(3,6)(4,5)(7,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,31,37)(26,33,32,40)(28,36,38,30)(29,35,34,39);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28)(26,34)(27,36)(29,32)(30,37)(31,38)(33,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29,31,34)(26,28,32,38)(27,35,37,39)(30,33,36,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30,31,36)(26,35,32,39)(27,28,37,38)(29,33,34,40)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015515);