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 | |
4791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,13)(12,16,20,22)(18,23,24,21)(25,26)(27,28)(29,37)(30,31)(32,35)(33,38)(34,39)(36,40);(1,2)(3,7)(4,5)(6,8)(10,15)(13,17)(16,22)(21,23)(26,31)(28,32)(29,38)(34,40);(2,5)(6,7)(12,20)(16,22)(18,24)(21,23)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,39,37,36);(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,16,15,22)(11,18,19,24)(13,21,17,23)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,11)(12,21)(14,17)(15,19)(16,18)(20,23)(22,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=gh,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9144235);