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) = C26 |
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 | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,14)(8,10)(9,13)(11,15)(12,18)(16,19)(17,20);(6,10)(8,18)(11,14)(12,13)(15,16)(19,20);(2,4)(5,6)(7,11)(8,12)(9,13)(10,18)(14,15)(16,20)(17,19);(5,7,9,17)(6,11,13,19)(8,16,18,15)(10,14,12,20)Polycyclic presentation:
2^8|a^2=f,d^2=h,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×#24457, #6437902, #6437904, … |
| maximal normal subgroup of | C2×#24457, #6437902, #6437904, … |
| maximal quotient of | #268632, #268635, #268646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17445);