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 | #40669 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,15,19,13)(12,20,23,16)(18,22,21,24)(26,31)(27,33)(28,35)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,21)(13,17)(14,19)(16,22)(18,23)(20,24)(25,26,30,31)(27,29,33,39)(28,36,35,32)(34,40,37,38);(2,3)(6,7)(9,12,14,23)(10,16,17,20)(11,18,19,21)(13,22,15,24)(25,27,30,33)(26,29,31,39)(28,37,35,34)(32,40,36,38);(4,8)(6,7)(9,13)(10,11)(12,22)(14,15)(16,18)(17,19)(20,21)(23,24)(25,26)(27,29)(28,32)(30,31)(33,39)(34,38)(35,36)(37,40);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,18)(13,22)(14,23)(15,24)(17,20)(19,21)(25,28,30,35)(26,32,31,36)(27,34,33,37)(29,38,39,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,24)(20,22)(21,23)(25,29)(26,33)(27,31)(28,38)(30,39)(32,37)(34,36)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8520090);