Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
471 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)(5,6,8,7)(9,10,14,15,13,16,12,18)(11,19,22,24,21,17,20,23);(5,7,8,6)(10,15)(11,20)(12,14)(16,18)(21,22)(23,24);(2,4)(10,16)(11,21)(15,18)(20,22);(9,11,12,20,13,21,14,22)(10,17,18,24,16,19,15,23)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23038, #6381944, #6381952, … |
| maximal normal subgroup of | C2×#23038, #6381944, #6381952, … |
| maximal quotient of | #229118, #229138, #229160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16026);