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 | |
4609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,29,31)(27,32,35,30)(28,36,37,33)(34,38,40,39);(2,6)(3,7)(12,19)(16,22)(18,23)(20,24)(25,27,29,35)(26,30,31,32)(28,34,37,40)(33,38,36,39);(1,2)(3,4)(5,6)(7,8)(10,15)(11,17)(16,22)(18,23)(26,31)(28,37)(30,32)(34,40);(2,6)(3,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,26)(27,30)(28,33)(29,31)(32,35)(34,38)(36,37)(39,40);(1,3,5,7)(2,4,6,8)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,27)(26,32)(28,34)(29,35)(30,31)(33,39)(36,38)(37,40);(1,4)(2,7)(3,6)(5,8)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,28)(26,33)(27,34)(29,37)(30,38)(31,36)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9344092);