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 | #45608 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,18)(12,20)(13,15)(14,16)(17,21)(19,23)(22,24)(26,31)(28,37)(29,36)(32,38);(2,5)(6,8)(10,13)(12,21)(16,18)(19,22)(26,31)(27,35)(28,37)(34,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,16,13,18)(12,19,21,22)(17,23,20,24)(25,26,30,31)(27,32,35,38)(28,33,37,39)(29,34,36,40);(9,12)(10,17)(11,19)(13,20)(14,22)(15,21)(16,23)(18,24)(25,27,30,35)(26,32,31,38)(28,36,37,29)(33,40,39,34);(1,2)(3,6)(4,5)(7,8)(9,13,15,10)(11,18,14,16)(12,20,21,17)(19,24,22,23)(25,28)(26,33)(27,36)(29,35)(30,37)(31,39)(32,40)(34,38);(1,3,4,7)(2,6,5,8)(9,14,15,11)(10,18,13,16)(12,22,21,19)(17,24,20,23)(25,29,30,36)(26,34,31,40)(27,28,35,37)(32,33,38,39)Polycyclic presentation:
2^9|c^2=h,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017761);