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 | |
4776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(26,30)(27,33)(32,36)(35,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,14)(16,24)(17,18)(19,22)(20,23)(25,26,29,30)(27,34,33,31)(28,36,38,32)(35,39,37,40);(11,13)(15,18)(19,21)(23,24)(25,27,29,33)(26,31,30,34)(28,37,38,35)(32,40,36,39);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,39)(36,40)(37,38);(9,10)(11,18)(12,16)(13,15)(14,17)(19,24)(20,22)(21,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,28,29,38)(26,32,30,36)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|b^2=h,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7714034);