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 | |
4567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,4,5)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,30,28)(27,32,35,31)(29,36,37,33)(34,38,40,39);(1,3)(2,7)(4,6)(5,8)(12,19)(16,22)(18,23)(20,24)(25,26)(27,31)(28,30)(29,33)(32,35)(34,38)(36,37)(39,40);(2,8)(3,4)(10,15)(11,17)(16,22)(18,23)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(1,3,6,4)(2,5,8,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,4,6,3)(2,5,8,7)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,28,30,26)(27,31,35,32)(29,33,37,36)(34,39,40,38);(1,5)(2,4)(3,8)(6,7)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=i,e^2=h,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, 9944866);