Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
527 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,13,10,14,9,12)(7,15,18,16,19,20,17,11);(1,2)(3,4)(5,7,8,18,10,19,9,17)(6,11,13,15,14,16,12,20);(2,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,18)(16,17)(19,20);(5,6)(7,16)(8,13)(9,12)(10,14)(11,19)(15,17)(18,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^c=fh,g^c=gh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25192, #6439024, #6439025, … |
| maximal normal subgroup of | C2×#25192, #6439024, #6439025, … |
| maximal quotient of | #283707, #283709, #283715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18180);