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 |
| prime radical | #42422 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4629 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)(3,7)(4,5)(6,8)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(2,5)(6,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(12,21)(16,20)(18,24)(22,23)(25,26,29,33)(27,31,35,30)(28,32,38,36)(34,40,37,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,26)(27,31)(28,36)(29,33)(30,35)(32,38)(34,39)(37,40);(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,22)(21,24)(25,27)(26,31)(28,37)(29,35)(30,33)(32,39)(34,38)(36,40);(9,13,14,10)(11,17,19,15)(12,16,21,20)(18,22,24,23)(25,28,29,38)(26,32,33,36)(27,34,35,37)(30,39,31,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8629412);