Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,15)(8,17)(9,13)(10,19)(11,14)(12,16)(18,20);(1,2,4,3)(5,7,9,16)(6,12,13,15)(8,11,18,10)(14,17,19,20);(5,8,10,16,9,18,11,7)(6,12,14,17,13,15,19,20);(1,3,4,2)(6,13)(12,15)(14,19)(17,20)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25748, #6443305, #6443313, … |
| maximal normal subgroup of | C2×#25748, #6443305, #6443313, … |
| maximal quotient of | #297359, #297360, #297369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18736);