Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,19)(18,20);(2,4)(6,10)(7,8)(11,12)(13,14)(15,17)(16,19)(18,20);(6,11)(7,8)(10,12)(14,17)(15,20)(18,19);(6,12)(10,11)Polycyclic presentation:
2^9|e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224710, #72224718, #72310284, … |
| maximal normal subgroup of | #72224710, #72224718, #72310284, … |
| maximal quotient of | C2×#233367 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140563);