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 |
| prime radical | #35102 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4616 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)(11,16)(12,18)(13,23)(14,20)(15,17)(19,21)(22,24)(25,26,30,34)(27,35,37,31)(28,33,29,32)(36,39,38,40);(3,7)(6,8)(9,11,15,20)(10,16,17,14)(12,21,22,13)(18,19,24,23)(25,27)(26,31)(28,36)(29,38)(30,37)(32,39)(33,40)(34,35);(1,2)(3,6)(4,5)(7,8)(10,17)(11,20)(13,21)(18,24)(25,28,30,29)(26,32,34,33)(27,36,37,38)(31,39,35,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,18)(11,13)(14,19)(15,22)(16,23)(17,24)(20,21)(25,29,30,28)(26,33,34,32)(27,36,37,38)(31,39,35,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,19)(11,12)(14,18)(15,21)(16,24)(17,23)(20,22)(25,27)(26,31)(28,38)(29,36)(30,37)(32,40)(33,39)(34,35);(9,14)(10,11)(12,19)(13,18)(15,16)(17,20)(21,24)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8023969);