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 |
| prime radical | #52335 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,12,16,15)(13,17,22,24)(19,20,23,21)(25,26)(27,36)(28,29)(30,40)(31,33)(32,37)(34,39)(35,38);(1,2)(3,7)(4,6)(5,8)(9,11,14,16)(10,15,18,12)(13,19,22,23)(17,21,24,20)(25,27,31,37)(26,32,33,36)(28,38,39,30)(29,40,34,35);(1,3)(2,5)(4,8)(6,7)(26,33)(27,37)(28,39)(35,40);(2,6)(5,7)(9,11)(10,15)(12,18)(13,19)(14,16)(17,21)(20,24)(22,23)(25,28)(26,34)(27,38)(29,33)(30,37)(31,39)(32,35)(36,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,15)(10,16,18,11)(13,21,22,20)(17,19,24,23)(25,29,31,34)(26,28,33,39)(27,35,37,40)(30,36,38,32);(1,3)(2,7)(4,8)(5,6)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,30,31,38)(26,35,33,40)(27,28,37,39)(29,36,34,32)Polycyclic presentation:
2^9|a^2=gh,b^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10126996);