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 | |
4540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(10,15)(11,17)(13,21)(20,23)(28,36)(31,37)(34,40)(38,39);(1,3,6,5)(2,7,4,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26)(27,32)(28,37)(29,33)(30,35)(31,36)(34,38)(39,40);(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,24)(22,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,40)(37,39);(1,2,6,4)(3,8,5,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,28,29,36)(26,31,33,37)(27,34,35,40)(30,38,32,39);(1,4,6,2)(3,8,5,7)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(1,5,6,3)(2,7,4,8)(9,13,14,21)(10,18,15,22)(11,19,17,24)(12,20,16,23)(25,26)(27,30)(28,31)(29,33)(32,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10388072);