Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,12,16,13,17,20,15)(11,14,18,22,19,23,24,21);(5,6)(7,8)(9,10)(11,14)(12,15)(13,17)(16,20)(18,21)(19,23)(22,24);(2,4)(6,8)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22);(2,4)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)Polycyclic presentation:
2^8|a^2=e,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6632132, #6632135, #6632164, … |
| maximal normal subgroup of | #6632132, #6632135, #6632164, … |
| maximal quotient of | #333275, #333290, #333304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23434);