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 | |
4645 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)(4,6)(5,8)(9,10)(11,15)(12,17)(13,21)(14,16)(18,22)(19,20)(23,24)(25,26,30,34)(27,31,35,40)(28,32,38,37)(29,33,39,36);(1,2,4,6)(3,5,8,7)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,23,21,20)(27,35)(28,38)(31,40)(32,37);(1,2)(3,5)(4,6)(7,8)(10,16)(11,17)(18,21)(19,24)(25,27)(26,31)(28,36)(29,32)(30,35)(33,38)(34,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,15)(10,17,16,11)(13,20,22,23)(18,24,21,19)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(9,13,14,22)(10,18,16,21)(11,19,17,24)(12,20,15,23);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22)(25,29)(26,33)(27,37)(28,31)(30,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8086838);