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 | |
4343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,13)(11,17,19,15)(12,16,21,22)(18,24,20,23)(25,26)(27,32)(28,35)(29,37)(30,31)(33,39)(34,36)(38,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,19)(10,15,13,17)(12,18,21,20)(16,23,22,24)(25,27,30,35)(26,28,31,32)(29,34,39,38)(33,36,37,40);(1,3)(2,5)(4,8)(6,7)(26,31)(28,32)(33,37)(36,40);(2,6)(5,7)(9,11)(10,15)(12,20)(13,17)(14,19)(16,24)(18,21)(22,23)(25,28)(26,27)(29,36)(30,32)(31,35)(33,34)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,22)(14,21)(15,23)(17,24)(19,20)(25,27)(26,32)(28,31)(29,38)(30,35)(33,36)(34,39)(37,40);(1,3)(2,7)(4,8)(5,6)(9,13,14,10)(11,17,19,15)(12,16,21,22)(18,23,20,24)(25,29)(26,33)(27,34)(28,36)(30,39)(31,37)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,f^2=gh,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, 10127012);