C₂×#1759 #20403

order	256
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#1759

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂³	C₂×C₂²⋊C₄
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

831 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(3,4,6,5)(7,8)(9,15)(10,18)(11,14)(12,17)(13,19)(16,21)(20,22);(7,9)(8,12,14,20)(10,17)(11,16)(13,15,18,21)(19,22);(3,5,6,4)(9,16)(12,20)(15,21)(17,22)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#105684, #105685, #106428, …
maximal normal subgroup of	#105684, #105685, #106428, …
maximal quotient of	C₂×#8170, C₂×#8171, #182145, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13391);

C2×#1759 #20403

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#1759 #20403