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 | |
4531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(25,26,30,32)(27,31,34,28)(29,37,40,33)(35,36,38,39);(1,3,4,7)(2,5,6,8)(10,15)(12,22)(16,18)(19,23)(25,27)(26,31)(28,32)(29,38)(30,34)(33,36)(35,40)(37,39);(1,2)(3,5)(4,6)(7,8)(10,15)(13,17)(16,18)(20,24)(26,32)(27,34)(33,37)(35,38);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,16,15,18)(12,23,22,19)(13,20,17,24)(25,28,30,31)(26,27,32,34)(29,39,40,36)(33,38,37,35);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,24)(21,23)(25,29)(26,33)(27,35)(28,36)(30,40)(31,39)(32,37)(34,38);(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,23,18,19)(25,27)(26,31)(28,32)(29,35)(30,34)(33,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300451);