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 | |
4695 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)(3,6)(4,8)(5,7)(9,10,14,11)(12,15,19,17)(13,16,22,18)(20,23,21,24)(27,33)(31,34)(35,39)(38,40);(1,2,5,7)(3,8,4,6)(10,11)(15,17)(16,18)(23,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,2,5,7)(3,8,4,6)(12,19)(15,17)(20,21)(23,24)(26,30)(31,34)(32,36)(38,40);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,17,19,15)(13,18,22,16)(20,24,21,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,2)(3,6)(4,8)(5,7)(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(1,4,5,3)(2,6,7,8)(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9928015);