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 | |
4474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,21)(13,17)(15,19)(16,18)(20,24)(22,23)(28,36)(31,37)(34,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,20)(13,23)(14,19)(15,18)(17,24)(21,22)(25,26)(27,32)(28,37)(29,33)(30,35)(31,36)(34,38)(39,40);(9,12,15,13)(10,17,19,21)(11,20,18,23)(14,22,16,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,40)(37,39);(1,2,5,4)(3,7,8,6)(9,13,15,12)(10,17,19,21)(11,20,18,23)(14,24,16,22)(25,28,29,36)(26,31,33,37)(27,34,35,40)(30,38,32,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,20)(13,23)(15,18)(16,19)(17,22)(21,24)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(1,4,5,2)(3,7,8,6)(9,14)(10,18)(11,19)(12,22)(13,24)(15,16)(17,23)(20,21)(25,26)(27,30)(28,31)(29,33)(32,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=gi,f^2=gh,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, 10389669);