Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10,14,18,13,17,20,23)(11,19,21,16,12,15,22,24);(1,2,4,3)(5,6)(7,8)(9,11,20,22,13,12,14,21)(10,15,23,16,17,19,18,24);(6,8)(9,12,13,11)(10,16,17,24)(14,21,20,22)(15,18,19,23);(1,3,4,2)(10,17)(15,19)(16,24)(18,23)Polycyclic presentation:
2^8|a^2=f,b^2=ef,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25389, #6444337, #6444355, … |
| maximal normal subgroup of | C2×#25389, #6444337, #6444355, … |
| maximal quotient of | #287744, #287750, #287756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18377);