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 | |
4553 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)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(25,26,31,33)(27,36,37,32)(28,38,35,30)(29,34,39,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,17,13)(14,19,23,20)(18,24,22,21)(25,27)(26,32)(28,29)(30,34)(31,37)(33,36)(35,39)(38,40);(1,2,4,6)(3,5,7,8)(10,17)(11,12)(14,23)(21,24)(26,33)(27,37)(28,35)(34,40);(3,7)(5,8)(9,12,15,11)(10,13,17,16)(14,20,23,19)(18,21,22,24)(25,28,31,35)(26,30,33,38)(27,29,37,39)(32,34,36,40);(1,3,4,7)(2,5,6,8)(9,13)(10,12)(11,17)(14,21)(15,16)(18,20)(19,22)(23,24)(25,29)(26,34)(27,35)(28,37)(30,36)(31,39)(32,38)(33,40);(1,2)(3,5)(4,6)(7,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,24)(17,22)(25,30)(26,35)(27,34)(28,33)(29,36)(31,38)(32,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219279, #72219281 |
| maximal quotient of | #72219276 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9361695);