Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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 | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
500 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)(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,24)(18,21)(19,22)(20,23);(1,2,4,3)(10,14)(11,17)(13,19)(15,23)(18,20)(21,24);(6,8)(9,11,12,18)(10,15,14,21)(13,20,19,17)(16,23,22,24);(1,3,4,2)(10,14)(15,21)(16,22)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=eh,c^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24737, #6442804, #6442819, … |
| maximal normal subgroup of | C2×#24737, #6442804, #6442819, … |
| maximal quotient of | #277027, #277028, #277029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17725);