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 | |
4320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(11,17)(15,18)(19,23)(22,24)(25,26)(27,30)(28,37)(29,33)(31,35)(32,38)(34,39)(36,40);(2,6)(3,7)(9,10,14,12)(11,18,17,15)(13,16,21,20)(19,24,23,22)(25,27,31,34)(26,30,35,39)(28,29,38,36)(32,40,37,33);(2,6)(3,7)(10,12)(15,18)(16,20)(22,24)(25,28,31,38)(26,32,35,37)(27,36,34,29)(30,33,39,40);(1,3,5,7)(2,4,6,8)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,28)(26,32)(27,36)(29,34)(30,33)(31,38)(35,37)(39,40);(1,2)(3,4)(5,6)(7,8)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,29,31,36)(26,33,35,40)(27,28,34,38)(30,37,39,32);(1,4)(2,7)(3,6)(5,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30)(26,34)(27,35)(28,33)(29,32)(31,39)(36,37)(38,40)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=hi,e^2=i,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, 9976291);