Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14,18,15)(16,17,20,19);(5,7)(6,9)(8,11)(10,12)(13,15,17,20,18,14,19,16);(2,4)(7,11)(9,12);(13,16,17,15,18,20,19,14)Polycyclic presentation:
2^8|a^2=gh,b^2=e,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27669, #6457156, #6457166, … |
| maximal normal subgroup of | C2×#27669, #6457156, #6457166, … |
| maximal quotient of | #314766, #314768, #314770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20657);