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 | |
4521 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,19,21,16)(12,20,23,18)(13,24,22,14)(25,26)(27,28)(29,37)(30,31)(32,39)(33,36)(34,38)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,21)(10,16,17,19)(12,22,23,13)(14,20,24,18)(25,27,30,33)(26,28,31,36)(29,34,39,40)(32,35,37,38);(2,5)(6,8)(10,17)(14,24)(16,19)(18,20)(26,31)(27,33)(32,37)(34,40);(3,7)(6,8)(9,12,15,23)(10,18,17,20)(11,13,21,22)(14,16,24,19)(25,28)(26,27)(29,38)(30,36)(31,33)(32,40)(34,37)(35,39);(1,3)(2,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,22)(16,20)(17,24)(18,19)(21,23)(25,29,30,39)(26,32,31,37)(27,34,33,40)(28,35,36,38);(1,2)(3,6)(4,5)(7,8)(9,14)(10,13)(11,20)(12,16)(15,24)(17,22)(18,21)(19,23)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8180672);