Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| 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 | #3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,9,13)(7,14,16,20)(8,10,18,12)(11,17,19,15);(1,3,4,2)(6,10)(8,18)(11,14)(12,13)(15,17)(19,20);(2,3)(5,6)(7,11)(8,12)(9,13)(10,18)(14,15)(16,19)(17,20);(5,7)(6,11)(8,17)(9,16)(10,14)(12,20)(13,19)(15,18)Polycyclic presentation:
2^8|a^2=f,b^2=fh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25184, #6438852, #6438854, … |
| maximal normal subgroup of | C2×#25184, #6438852, #6438854, … |
| maximal quotient of | #282571, #282574, #282583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18172);