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 |
| prime radical | #3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,11,14,12,9,10,15)(7,16,17,13,8,19,18,20);(1,2)(3,4)(5,7,11,17,12,8,10,18)(6,13,14,19,9,20,15,16);(2,4)(5,8,12,7)(6,13,9,20)(10,18,11,17)(14,16,15,19);(5,9,12,6)(7,13,8,20)(10,14,11,15)(16,17,19,18)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=h,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×#25195, #6439024, #6439025, … |
| maximal normal subgroup of | C2×#25195, #6439024, #6439025, … |
| maximal quotient of | #283712, #283714, #283720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18183);